# Boolean Algebras. Chapter 2

Save this PDF as:

Size: px
Start display at page:

## Transcription

1

2 Chapter 2 Boolean Algebras Let X be an arbitrary set and let P(X) be the class of all subsets of X (the power set of X). Three natural set-theoretic operations on P(X) are the binary operations of union and intersection, and the unary operation of complementation. The union P Q of two subsets P and Q is, by definition, the set of elements that are either in P or in Q, the intersection P Q is the set of elements that are in both P and Q, and the complement P is the set of elements (of X) that are not in P. There are also two distinguished subsets: the empty set, which has no elements, and the universal set X. The class P(X), together with the operations of union, intersection, and complementation, and the distinguished subsets and X, is called the Boolean algebra (or field) of all subsets of X, or the power set algebra on X. The arithmetic of this algebra bears a striking resemblance to the arithmetic of Boolean rings. Some of the most familiar and useful identities include the laws for forming the complements of the empty and the universal sets, (1) = X, X =, the laws for forming an intersection with the empty set and a union with the universal set, (2) P =, P X = X, the identity laws, (3) P X = P, P = P, the complement laws, (4) P P =, P P = X, S. Givant, P. Halmos, Introduction to Boolean Algebras, 8 Undergraduate Texts in Mathematics, DOI: / , c Springer Science+Business Media, LLC 2009

3 2 Boolean Algebras 9 the double complement law, (5) (P ) = P, the idempotent law, (6) P P = P, P P = P, the De Morgan laws, (7) (P Q) = P Q, (P Q) = P Q, the commutative laws, (8) P Q = Q P, P Q = Q P, the associative laws, (9) P (Q R) =(P Q) R, P (Q R) =(P Q) R, and the distributive laws, (10) P (Q R) =(P Q) (P R), P (Q R) =(P Q) (P R). Each of these identities can be verified by an easy set-theoretic argument based on the definitions of the operations involved. Consider, for example, the verification of the first De Morgan law. It must be shown that each element x of X belongs to (P Q) just in case it belongs to P Q. The argument goes as follows: x (P Q) if and only if x P Q, if and only if x P or x Q, if and only if x P or x Q, if and only if x P Q. The first and third equivalences use the definition of complementation, the second uses the definition of intersection, and the last uses the definition of union. While (1) (10) bear a close resemblance to laws that are true in Boolean rings, there are important differences. Negation in Boolean rings is the identity operation, whereas complementation is not. Addition in Boolean rings is not an idempotent operation, whereas union is. The distributive law for addition over multiplication fails in Boolean rings, whereas the distributive law for union over intersections holds.

4 10 Introduction to Boolean Algebras Boolean rings are an abstraction of the ring 2. The corresponding abstraction of P(X) is called a Boolean algebra. Specifically, a Boolean algebra is a non-empty set A, together with two binary operations and (on A), a unary operation, and two distinguished elements 0 and 1, satisfying the following axioms, the analogues of identities (1) (10): (11) (12) (13) (14) (15) (16) 0 =1, 1 =0, p 0=0, p 1=1, p 1=p, p 0=p, p p =0, p p =1, (p ) = p, p p = p, p p = p, (17) (18) (19) (20) (p q) = p q, (p q) = p q, p q = q p, p q = q p, p (q r) =(p q) r, p (q r) =(p q) r, p (q r) =(p q) (p r), p (q r) =(p q) (p r). This set of axioms is wastefully large, more than strong enough for the purpose. The problem of selecting small subsets of this set of conditions that are strong enough to imply them all is one of dull axiomatics. For the sake of the record: one solution of the problem, essentially due to Huntington [28], is given by the identity laws (13), the complement laws (14), the commutative laws (18), and the distributive laws (20). To prove that these four pairs imply all the other conditions, and, in particular, to prove that they imply the De Morgan laws (17) and the associative laws (19), involves some non-trivial trickery. There are several possible widely adopted names for the operations,, and. We shall call them meet, join, andcomplement (or complementation), respectively. The distinguished elements 0 and 1 are called zero and one. One is also known as the unit. Equations (1) (10) imply that the class of all subsets of an arbitrary set X is an example of a Boolean algebra. When the underlying set X is empty, the resulting algebra is degenerate in the sense that it has just one element. In this case, the operations of join, meet, and complementation are all constant, and 0 = 1. The simplest non-degenerate Boolean algebra is the class of all subsets of a one-element set. It has just two elements, 0 (the empty set)

5 2 Boolean Algebras 11 and 1 (the one-element set). The operations of join and meet are described by the arithmetic tables and and complementation is the unary operation that maps 0 to 1, and conversely. We shall see in a moment that this algebra and the two-element Boolean ring are interdefinable. For that reason, the same symbol 2 is used to denote both structures. Here is a comment on notation, inspired by the associative laws (19). It is an elementary consequence of those laws that if p 1,...,p n are elements of a Boolean algebra, then p 1 p n, makes sense. The point is, of course, that since such joins are independent of how they are bracketed, it is not necessary to indicate any bracketing at all. The element p 1 p n may alternatively be denoted by n i=1 p i, or, in case no confusion is possible, simply by i p i. If we make simultaneous use of both the commutative and the associative laws, we can derive a slight but useful generalization of the preceding comment. If E is a non-empty finite subset of a Boolean algebra, then the set E has a uniquely determined join, independent of any order or bracketing that may be used in writing it down. (In case E is a singleton, it is natural to identify that join with the unique element in E.) We shall denote the join of E by E. Both the preceding comments apply to meets as well as to joins. The corresponding symbols are, of course, n p i, or p i, and E. i=1 i The conventions regarding the order of performing different operations in the absence of any brackets are the following: complements take priority over meets and joins, while meets take priority over joins. Example: the expression p q p should be read as (p ) (q p). It is convenient to write successive applications of complement without any bracketing, for instance p instead of (p ). Exercises 1. Verify that the identities (1) (10) are true in every Boolean algebra of all subsets of a set.,

6 12 Introduction to Boolean Algebras 2. (Harder.) Show that the identities in (13), (14), (18), and (20) together form a set of axioms for the theory of Boolean algebras. In other words, show that they imply the identities in (11), (12), (15), (16), (17), and (19). (This result is essentially due to Huntington [30].) 3. Prove directly that the two-element structure 2 defined in the chapter is a Boolean algebra, by showing that axioms (13), (14), (18), and (20) are all valid in In analogy with the construction, for each set X, of the Boolean ring 2 X in Chapter 1, define operations of join, meet, and complementation on 2 X, and distinguished constants zero and one, and prove that the resulting structure is a Boolean algebra. 5. (Harder.) A member of a set of axioms is said to be independent of the remaining axioms if it is not derivable from them. One technique for demonstrating the independence of a given axiom is to construct a model in which that axiom fails while the remaining axioms hold. The given axiom cannot then be derivable from the remaining ones, since if it were, it would have to hold in the model as well. The four pairs of identities (13), (14), (18), and (20) constitute a set of eight axioms for Boolean algebras. (a) Show that the distributive law for join over meet in (20) is independent of the remaining seven axioms. (b) Show that the distributive law for meet over join in (20) is independent of the remaining seven axioms. (c) Show that each of the complement laws in (14) is independent of the remaining seven axioms. (These proofs of independence are due to Huntington [28].) 6. (Harder.) A set of axioms is said to be independent if no one of the axioms can be derived from the remaining ones. Do the four pairs of identities (13), (14), (18), and (20) constitute an independent set of axioms for Boolean algebras? 7. (Harder.) The operation of meet and the distinguished elements zero and one can be defined in terms of join and complement by the equations

7 2 Boolean Algebras 13 p q =(p q ), 0=(p p ), 1=p p. A Boolean algebra may therefore be thought of as a non-empty set together with two operations: join and complement. Prove that the following identities constitute a set of axioms for this conception of Boolean algebras: the commutative and associative laws for join, and (H) (p q ) (p q) = p. (This axiomatization, and the proof of its equivalence to the set of axioms (13), (14), (18), and (20), is due to Huntington [30]. In fact, (H) is often called Huntington s axiom.) 8. (Harder) Prove that the three axioms in Exercise 7 are independent. (The proof of independence is due to Huntington [30].) 9. (Harder.) Prove that the following identities constitute a set of axioms for Boolean algebras: p = p, p (q q ) = p, p (q r) =((q p) (r p) ). (This axiomatization, and the proof of its equivalence with the axiom set in Exercise 7, is due to Huntington [30].) 10. (Harder.) Prove that the three axioms in Exercise 9 are independent. (The proof of independence is due to Huntington [30].) 11. (Harder.) Prove that the commutative and associative laws for join, and the equivalence p q = r r if and only if p q = p, together constitute a set of axioms for Boolean algebras. (This axiomatization, and the proof of its equivalence with the axiom set in Exercise 7, is due to Byrne [11].)

8 Chapter 5 Fields of Sets To form P(X) is not the only natural way to make a Boolean algebra out of a non-empty set X. A more general way is to consider an arbitrary non-empty subclass A of P(X) that is closed under intersection, union, and complement; in other words, if P and Q are in A, then so are P Q, P Q, andp. Since A contains at least one element, it follows that A contains and X (cf. (2.4)), and hence that A is a Boolean algebra. Every Boolean algebra obtained in this way is called a field (of sets). There is usually no danger in denoting a field of sets by the same symbol as the class of sets that go to make it up. This does not, however, justify the conclusion (it is false) that set-theoretic intersection, union, and complement are the only possible operations that convert a class of sets into a Boolean algebra. To show that a class A of subsets of a set X is a field, it suffices to show that A is non-empty and closed under union and complement. In fact, if A is closed under these two operations, then it must also be closed under intersection, since P Q =(P Q ) for any two subsets P and Q of X. Dually, A is a field whenever it is nonempty and closed under intersection and complement. A subset P of a set X is cofinite (in X) if its complement P is finite; in other words, P is cofinite if it can be obtained from X by removing finitely many elements. For instance, the set of integers greater than 1000 or less than 10 is cofinite (in the set of all integers), and so is the set of all integers with the numbers 2, 75, and 1037 removed; the set of even integers is not cofinite. The class A of all those subsets of a non-empty set X that are either finite or cofinite is a field of subsets of X. The proof is a simple cardinality S. Givant, P. Halmos, Introduction to Boolean Algebras, 24 Undergraduate Texts in Mathematics, DOI: / , c Springer Science+Business Media, LLC 2009

9 5 Fields of Sets 25 argument. The union of two finite subsets is finite, while the union of a cofinite subset with any subset is cofinite; also, the complement of a finite subset is cofinite, and conversely. If the set X itself is finite, then A is simply P(X); if X is infinite, then A is a new example of a Boolean algebra and is called the finite cofinite algebra (or field) ofx. The preceding construction can be generalized. Call a subset P of X cocountable (in X) if its complement P is countable. For example, the set of irrational numbers is cocountable in the set of real numbers, since the set of rational numbers is countable. Also, any cofinite subset of reals is cocountable, since any finite subset is countable. The class of all those subsets of X that are either countable or cocountable is a field of subsets of X, the so-called countable cocountable algebra (or field) ofx. Different description of the same field: the class of all those subsets P of X for which the cardinal number of either P or P is less than or equal to ℵ 0 (the first infinite cardinal). A further generalization is obtained by using an arbitrary infinite cardinal number in place of ℵ 0. Let X be the set of all integers (positive, negative, or zero), and let m be an arbitrary integer. A subset P of X is periodic of period m if it coincides with the set obtained by adding m to each of its elements. The class A of all periodic sets of period m is a field of subsets of X. If m = 0, then A is simply P(X). If m = 1, then A consists of just the two sets and X. Inall other cases A is a new example of a Boolean algebra. For example, if m =3, then A consists of eight sets, namely the eight possible unions that can be formed using one, two, three, or none of the sets {3n : n X}, {3n +1:n X}, and {3n +2:n X}. It is obvious in this case that A is closed under the operation of union, and almost obvious that A is closed under complementation. Consequently, A is a field of sets. Warning: a period of a periodic set is not unique; for example, the sets of integers of period 3 also have period 6 and period 12; in fact they have infinitely many periods. Let X be the set of all real numbers. A left half-closed interval (or, for brevity, since this is the only kind we shall consider for the moment, a half-closed interval) is a set of one of the forms [a, b) ={x X : a x<b}, [a, ) ={x X : a x}, (,b)={x X : x<b}, (, ) =X,

10 26 Introduction to Boolean Algebras where, of course, a and b themselves are real numbers and a<b. The class A of all finite unions of half-closed intervals is a field of subsets of X. Here is the proof. The empty set is the union of the empty family of half-closed intervals, so it belongs to A. The closure of A under union is obvious: the union of two finite unions of half-closed intervals is itself a finite union of half-closed intervals. To establish the closure of A under complement, it is helpful to make two observations. First, the intersection of two half-closed intervals is either a half-closed interval or empty. For instance, if a 1 <b 2 and a 2 <b 1, then where [a 1,b 1 ) [a 2,b 2 )=[c, d) and [a 1,b 1 ) (,b 2 )=[a 1,d), c = max{a 1,a 2 } and d =min{b 1,b 2 }. (The diagram illustrates the three possibilities in this case.) The intersection a b a b a b a b a c d c d c d b a b of the intervals is empty if the relevant inequalities fail. The second observation is that the complement of a half-closed interval is the union of at most two half-closed intervals. For instance, the complement of the interval [a, b) is the union (,a) [b, + ), while the complement of the interval [a, + ) is(,a). If P is a finite union of half-closed intervals, then P is a finite intersection of complements of half-closed intervals, by the De Morgan laws (2.7). Each of these complements is the union of at most two half-closed intervals, by the second observation, so P is the intersection of a finite family of sets, each of which is the union of at most two half-closed intervals. The distributive laws (2.10) therefore imply that P may be written as a finite union of finite intersections of half-closed intervals. Each of the latter intersections is again a half-closed interval or else empty, by the first observation, so P may be

11 5 Fields of Sets 27 written as a finite union of half-closed intervals. The class A is thus closed under complementation. Any (linearly) ordered set can be used instead of the set of real numbers, though some details of the construction may require slight modification. For instance, a useful variant uses the closed unit interval [0, 1] in the role of X. In this case it is convenient to stretch the terminology so as to include the closed intervals [a, 1] and the degenerate interval {1} among half-closed intervals. (The elements of this field are just the intersections with [0, 1] of the finite unions of half-closed intervals discussed in the preceding paragraph. Notice, in this connection, that [a, 1]=[a, b) [0, 1] and {1} =[1,b) [0, 1] whenever 0 a 1 <b.) The field of sets constructed from an ordered set in the manner illustrated by these two examples is usually called the interval algebra of the ordered set. (Interval algebras were first introduced by Mostowski and Tarski in [46].) Valuable examples of fields of sets can be defined in the unit square (the set of ordered pairs (x, y) with0 x, y 1), as follows. Call a subset P of the square vertical if along with each point in P, every point of the vertical segment through that point also belongs to P. In other words, P is vertical if the presence of (x 0,y 0 )inp implies the presence in P of (x 0,y) for every y in [0, 1]. If A is any field of subsets of the square, then the class of all vertical sets in A is another, and in particular the class of all vertical sets is a field of sets. Indeed, A contains the empty set, and is closed under union and complement, so it suffices to check that the empty set is a vertical set (this is vacuously true), and that the union of two vertical sets and the complement of a vertical set are again vertical sets. Let P and Q be vertical sets, and suppose that a point (x 0,y 0 ) belongs to their union. If the point belongs to P, then every point on the vertical segment through (x 0,y 0 )also belongs to P (since P is vertical); consequently, every such point belongs to the union P Q. An analogous argument applies if (x 0,y 0 ) belongs to Q. It follows that P Q is a vertical set. Suppose, next, that a point (x 0,y 0 ) belongs to the complement of P. No point on the vertical segment through (x 0,y 0 ) can then belong to P, for the presence of one such point in P implies the presence of every such point in P (since P is vertical), and in particular it implies that (x 0,y 0 )isinp. It follows that every point on the vertical segment through (x 0,y 0 ) belongs to P, so that P is a vertical set. Here are two comments that are trivial but sometimes useful: (1) the horizontal sets (whose definition may safely be left to the reader) constitute

12 28 Introduction to Boolean Algebras just as good a field as the vertical sets, and (2) the Cartesian product of any two non-empty sets is, for these purposes, just as good as the unit square. Exercises 1. Prove that the class of all sets of integers that are either finite sets of even integers, or else cofinite sets that contain all odd integers, is a field of sets. 2. Prove that the class of all sets of real numbers that are either countable or cocountable is a field of sets. 3. Prove that the class of all periodic sets of integers of period 2 is a field of sets. How many sets are in this class? Describe them all. Do the same for the class of all period sets of integers of period Prove that the class of all periodic sets of integers of period m is a field of sets. How many sets are in this class? Describe them all. 5. (Harder.) If m is a positive integer, and if A is the class of all those sets of integers that are periodic of some period greater than m, isa a field of sets? 6. The precise formulation and proof of the assertion that the intersection of two half-closed intervals is either half-closed or empty can be simplified by introducing some unusual but useful notation. Extend the natural linear ordering of the set X of real numbers to the set X {, } of extended real numbers by requiring <a< for every real number a, and <. Thus, for example, max{,a} = a and min{,a} =. Extend the notation for half-closed intervals by defining [,a)=(,a)={b X : b<a} for any a in X { }. In particular, [, ) =(, ) =X. Also, extend the notation [a, b) to the case a b by writing

13 5 Fields of Sets 29 [a, b) ={x R : a x<b} =. The notation [a, b) thus makes sense for any two extended real numbers a and b. The assertion that the intersection of two half-closed intervals is either half-closed or empty can be expressed in this notation by the following simple statement that avoids case distinctions: where [a 1,b 1 ) [a 2,b 2 )=[c, d), c = max{a 1,a 2 } and d =min{b 1,b 2 }; moreover, this intersection is non-empty if and only if a 1,a 2 <b 1,b 2. Prove the simple statement. 7. This exercise continues with the notation for extended real numbers introduced in Exercise 6. Prove that every set P in the interval algebra A of the real numbers can be written in exactly one way in the form P =[a 1,b 1 ) [a 2,b 2 ) [a n,b n ), where n is a non-negative integer and a 1 <b 1 <a 2 <b 2 < <a n <b n. (The case n = 0 is to be interpreted as the empty family of half-closed intervals.) Show that if P has this form, then P =[,a 1 ) [b 1,a 2 ) [b n 1,a n ) [b n, + ); the first interval is empty (and hence should be omitted) if a 1 = ; the second interval is empty (and hence should be omitted) if b n =+. 8. Prove that the class of all finite unions of left half-closed intervals of rational numbers is a countable field of sets. 9. Prove that the class of all finite unions of left half-closed subintervals of the interval [0, 1] is a field of sets (where the intervals [a, 1], for 0 a 1, are considered left half-closed). 10. Let X be the set of real numbers. A right half-closed interval of real numbers is a set of one of the forms (,b]={x X : x b}, (a, b] ={x X : a<x b},

14 30 Introduction to Boolean Algebras (a, + ) ={x X : a<x}, (, + ) =X, where a and b are real numbers and a<b. Show that the class of all finite unions of right half-closed intervals is a field of subsets of the set of real numbers. 11. An interval of real numbers is a set P of real numbers with the property that whenever c and d are P, then so is every number between c and d. In other words, an interval is a set of one of the forms (,a), (,a], (a, + ), [a, + ), (, + ), (a, b), (a, b], [a, b), [a, b],, where a and b are real numbers and a<b. Prove that the class of all finite unions of intervals is a field of sets. 12. Define the notion of a horizontal set for the unit square, and prove that the class of all such sets is a field of sets. 13. Give an example of a Boolean algebra whose elements are subsets of a set, but whose operations are not the usual set-theoretic ones.

15 Chapter 7 Order We continue to work with an arbitrary but fixed Boolean algebra A. Lemma 1. p q = p ifandonlyifp q = q. Proof. If p q = p, then p q =(p q) q, and the conclusion follows from the appropriate law of absorption. The converse implication is obtained from this one by interchanging the roles of p and q and forming duals. For sets, either one of the equations P Q = P and P Q = Q is equivalent to the inclusion P Q. This observation motivates the introduction of a binary relation in every Boolean algebra; we write p q or q p in case p q = p, or, equivalently, p q = q. A convenient way of expressing the relation p q inwordsistosaythatp is below q, and that q is above p, or that q dominates p. The relation of inclusion between sets is a partial order. In other words, the relation is reflexive in the sense that the inclusion P P always holds; it is antisymmetric in the sense that the inclusions P Q and Q P imply P = Q; and it is transitive in the sense that the inclusions P Q and Q R imply P R. The next lemma says that the same thing is true of the relation in an arbitrary Boolean algebra. Lemma 2. The relation is a partial order. S. Givant, P. Halmos, Introduction to Boolean Algebras, 38 Undergraduate Texts in Mathematics, DOI: / , c Springer Science+Business Media, LLC 2009

16 7 Order 39 Proof. Reflexivity follows from the idempotent law (2.16): p p = p, and therefore p p. Antisymmetry follows from the commutative law (2.18): if p q and q p, then p = p q = q p = q. Transitivity follows from the associative law (2.19): if p q and q r, then and consequently p q = p and q r = q, p r =(p q) r = p (q r) =p q = p. It is sound mathematical practice to re-examine every part of a structure in the light of each new feature soon after the novelty is introduced. Here is the result of an examination of the structure of a Boolean algebra in the light of the properties of order. Lemma 3. (1) 0 p and p 1. (2) If p q and r s, then p r q s and p r q s. (3) If p q, then q p. (4) p q ifandonlyifp q =0,or, equivalently, p q =1. The proofs of all these assertions are automatic. It is equally automatic to discover the dual of ; according to any reasonable interpretation of the phrase it is. The inequalities in (2) are called the monotony laws.. If E is any subset of a partially ordered set such as our Boolean algebra A, we can consider the set F of all upper bounds of E and ask whether F has a smallest element. In other words: an element q belongs to F in case p q for every p in E; to say that F has a smallest element means that there exists an element q 1 in F such that q 1 q for every q in F. A smallest element in F, if it exists, is obviously unique: if q 1 and q 2 both have this property, then q 1 q 2 and q 2 q 1, so that q 1 = q 2, by antisymmetry. We shall call the smallest upper bound of the set E (if it has one) the least upper bound, or the supremum, of E. All these considerations have their obvious duals. The greatest lower bound of E is also called the infimum of E. If the set E is empty, then every element of A is vacuously an upper bound of E (p in E implies p q for each q in A), and, consequently, E

17 40 Introduction to Boolean Algebras has a supremum, namely 0. Similarly (dually), if E is empty, then E has an infimum, namely 1. Consider next the case of a singleton, say {p}. Since p itself is an upper bound of this set, it follows that the set has a supremum, namely p, and, similarly, that it has an infimum, namely p again. The situation becomes less trivial when we pass to sets of two elements. Lemma 4. For each p and q, the set {p, q} has the supremum p q and the infimum p q. Proof. The element p q dominates both p and q, by the absorption laws, so it is one of the upper bounds of {p, q}. It remains to prove that p q is the least upper bound, or, in other words, that if both p and q are dominated by some element r, then p q r. This is easy; by (2) and idempotence, p q r r = r. The assertion about the infimum follows by duality. Lemma 4 generalizes immediately to arbitrary non-empty finite sets (instead of sets with only two elements). We may therefore conclude that if E is a non-empty finite subset of A, then E has both a supremum and an infimum, namely E and E respectively. Motivated by these facts we hereby extend the interpretation of the symbols used for joins and meets to sets that may be empty or infinite. If a subset E of A has a supremum, we shall denote that supremum by E regardless of the size of E, and, similarly, we shall use E for all infima. In this notation what we know about very small sets can be expressed as follows: =0, =1, {p} = {p} = p. The notation used earlier for the join or meet of a finite sequence of elements is also extendable to the infinite case. Thus if {p i } is an infinite sequence with a supremum (properly speaking, if the range of the sequence has a supremum), then that supremum is denoted by i=1 p i. If, more generally, {p i } is an arbitrary family with a supremum, indexed by the elements i of a set I, the supremum is denoted by i I p i, or, in case no confusion is possible, simply by i p i. The perspective of Boolean algebras as partially ordered sets goes back to Jevons [31] and suggests a natural generalization. A lattice is a partially ordered set in which, for any elements p and q, the set {p, q} has both a supremum and an infimum. Two binary operations called join and meet

18 7 Order 41 may be introduced into a lattice as follows: the join of elements p and q, written p q, is defined to be the supremum of the set {p, q}, and the meet of p and q, written p q, is the infimum of {p, q}. The two operations satisfy the idempotent laws (2.16), the commutative laws (2.18), the associative laws (2.19), and the absorption laws (4.1). Consequently, the analogue of Lemma 1 also holds for lattices, and one can easily show that p q if and only if p q = p, or, equivalently, if and only if p q = p. There is an alternative approach to lattices: they may be defined as algebraic structures consisting of a non-empty set and two binary operations denoted by and that satisfy the idempotent, commutative, associative, and absorption laws. A binary relation may then be defined in exactly the same way as it is defined for Boolean algebras, and the analogues of Lemmas 2 and 4 may be proved. The situation may be summarized by saying that the two conceptions of a lattice are definitionally equivalent. Notice that the axioms for lattices in this alternative approach come in dual pairs. Consequently, there is a principle of duality for lattices that is the exact analogue of the principle of duality for Boolean algebras (Chapter 4). A lattice may or may not have a smallest or a greatest element. The smallest element, if it exists, is called the zero of the lattice and is usually denoted by 0; the largest element, if it exists, is called the one, or the unit, of the lattice and is denoted by 1. It is easy to check that the zero and unit of a lattice (if they exist) satisfy the identities in (2.12) and (2.13). The distributive laws in (2.20) may fail in a lattice; however, if one of them does hold, then so does the other, and the lattice is said to be distributive. A lattice in which every subset has an infimum and a supremum is said to be complete. The infimum and supremum of a set E, if they exist, are denoted by E and E respectively. A complete lattice automatically has a least and a greatest element, namely, the infimum and the supremum of the set of all elements in the lattice. Every Boolean algebra, under the operations of join and meet, is a lattice, and in fact a distributive lattice. The converse is not necessarily true. For instance, the set of natural numbers under the standard ordering relation is a distributive lattice, but it is not a Boolean algebra. The reason for mentioning lattices in this book is that they naturally arise in the study of Boolean algebras. We shall encounter some important examples later.

19 42 Introduction to Boolean Algebras Exercises 1. Prove that a Boolean algebra is degenerate if and only if 1 = Prove Lemma Prove the converse to part (3) of Lemma 3: if q p, then p q. 4. Prove that p q is the dual of p q. 5. If p q and p 0, prove that q p q. 6. The concept of divisibility makes sense in every ring: p is divisible by q in case p = q r for some r. Show that in a Boolean ring an element p is divisible by q if and only if p q. 7. True or false: if p q and r s, then p + r q + s and p r q s. 8. Prove that p r (p q) (q r), and equality holds if r q p. 9. Derive the following inequalities. (a) p + r (p + q) (q + r). (b) (p q)+(r s) (p + r) (q + s). 10. Derive the following inequalities. (a) (p q) (q r) p r. (b) (p r) (q s) (p q) (r s). 11. Derive the following inequalities. (a) (p q) (q r) p r. (b) (p r) (q s) (p q) (r s). 12. Give a precise definition of the notion of the infimum of a set E in a Boolean algebra. 13. Prove that in every lattice, conceived as a partially ordered set, the defined operations of join and meet satisfy the idempotent, commutative, associative, and absorption laws. Use these laws to show that Lemma 1 and part (2) of Lemma 3 hold for lattices and that p q if and only if p q = p.

20 7 Order Suppose lattices are conceived as algebraic structures with two binary operations satisfying the idempotent, commutative, associative, and absorption laws. Prove that the binary relation defined by p q if and only if p q = p is a partial order and the monotony laws (part (2) of Lemma 3) hold. Show further that Lemma 4 remains true. (Exercises 13 and 14 say, together, that the conceptions of a lattice as a partially ordered set, and as a structure with two binary operations satisfying the idempotent, commutative, associative, and absorption laws, are definitionally equivalent.) 15. Show that in a lattice with zero and one, the laws (2.12) and (2.13) are valid. 16. Show, conversely, that if a lattice has elements 0 and 1 satisfying the laws in (2.12), or else the laws in (2.13), then 0 is the least element, and 1 is the greatest element, in the lattice. (Exercises 15 and 16 say, together, that the two conceptions of a lattice with zero and one the first as a partially ordered set with a greatest and a least element, the second as a structure with two binary operations satisfying the idempotent, commutative, associative, absorption, and either (2.12) or the identity laws are definitionally equivalent.) 17. Prove that a part of each of the distributive laws, namely the inequalities (p q) (p r) p (q r) and (p q) (p r) p (q r), holds in every lattice. 1 p q r The preceding diagram determines a five-element lattice in which

21 44 Introduction to Boolean Algebras p q = p r = q r =1 and p q = p r = q r =0. Show that both distributive laws fail in this lattice. 19. Prove that in every lattice the validity of one of the two distributive laws p (q r) =(p q) (p r) and p (q r) =(p q) (p r) implies the validity of the other. 20. In a lattice with zero and one, a complement of an element p is an element q such that p q =0 and p q =1. Prove that in a distributive lattice with zero and one, a complement of an element, if it exists, is unique. 21. Show that any linearly ordered set with a least and a greatest element, and with more than two elements, is an example of a distributive lattice with zero and one that is not a Boolean algebra. 22. A lattice with zero and one is said to be complemented if every element has a complement (see Exercise 20). Interpret and prove the assertion that complemented distributive lattices are the same thing as Boolean algebras. 23. Prove that a lattice in which every set of elements has a least upper bound is complete.

### Exercises for Unit VI (Infinite constructions in set theory)

Exercises for Unit VI (Infinite constructions in set theory) VI.1 : Indexed families and set theoretic operations (Halmos, 4, 8 9; Lipschutz, 5.3 5.4) Lipschutz : 5.3 5.6, 5.29 5.32, 9.14 1. Generalize

### Lecture 4. Algebra, continued Section 2: Lattices and Boolean algebras

V. Borschev and B. Partee, September 21-26, 2006 p. 1 Lecture 4. Algebra, continued Section 2: Lattices and Boolean algebras CONTENTS 1. Lattices.... 1 1.0. Why lattices?... 1 1.1. Posets... 1 1.1.1. Upper

### Axioms of Kleene Algebra

Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.

### Infinite constructions in set theory

VI : Infinite constructions in set theory In elementary accounts of set theory, examples of finite collections of objects receive a great deal of attention for several reasons. For example, they provide

### Boolean Algebra CHAPTER 15

CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 1-1 and 4-1 (in Chapters 1 and 4, respectively). These laws are used to define an

### Logic via Algebra. Sam Chong Tay. A Senior Exercise in Mathematics Kenyon College November 29, 2012

Logic via Algebra Sam Chong Tay A Senior Exercise in Mathematics Kenyon College November 29, 2012 Abstract The purpose of this paper is to gain insight to mathematical logic through an algebraic perspective.

### Analysis I. Classroom Notes. H.-D. Alber

Analysis I Classroom Notes H-D Alber Contents 1 Fundamental notions 1 11 Sets 1 12 Product sets, relations 5 13 Composition of statements 7 14 Quantifiers, negation of statements 9 2 Real numbers 11 21

### Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

### Some Basic Notations Of Set Theory

Some Basic Notations Of Set Theory References There are some good books about set theory; we write them down. We wish the reader can get more. 1. Set Theory and Related Topics by Seymour Lipschutz. 2.

### Structure of R. Chapter Algebraic and Order Properties of R

Chapter Structure of R We will re-assemble calculus by first making assumptions about the real numbers. All subsequent results will be rigorously derived from these assumptions. Most of the assumptions

### Lecture Notes in Real Analysis Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay

Lecture Notes in Real Analysis 2010 Anant R. Shastri Department of Mathematics Indian Institute of Technology Bombay August 6, 2010 Lectures 1-3 (I-week) Lecture 1 Why real numbers? Example 1 Gaps in the

### Equational Logic. Chapter Syntax Terms and Term Algebras

Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from

### Real Analysis - Notes and After Notes Fall 2008

Real Analysis - Notes and After Notes Fall 2008 October 29, 2008 1 Introduction into proof August 20, 2008 First we will go through some simple proofs to learn how one writes a rigorous proof. Let start

### Analysis 1. Lecture Notes 2013/2014. The original version of these Notes was written by. Vitali Liskevich

Analysis 1 Lecture Notes 2013/2014 The original version of these Notes was written by Vitali Liskevich followed by minor adjustments by many Successors, and presently taught by Misha Rudnev University

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### 1. (B) The union of sets A and B is the set whose elements belong to at least one of A

1. (B) The union of sets A and B is the set whose elements belong to at least one of A or B. Thus, A B = { 2, 1, 0, 1, 2, 5}. 2. (A) The intersection of sets A and B is the set whose elements belong to

### Topics in Logic and Proofs

Chapter 2 Topics in Logic and Proofs Some mathematical statements carry a logical value of being true or false, while some do not. For example, the statement 4 + 5 = 9 is true, whereas the statement 2

### 1 Fields and vector spaces

1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary

### Congruence Boolean Lifting Property

Congruence Boolean Lifting Property George GEORGESCU and Claudia MUREŞAN University of Bucharest Faculty of Mathematics and Computer Science Academiei 14, RO 010014, Bucharest, Romania Emails: georgescu.capreni@yahoo.com;

### Footnotes to Linear Algebra (MA 540 fall 2013), T. Goodwillie, Bases

Footnotes to Linear Algebra (MA 540 fall 2013), T. Goodwillie, Bases November 18, 2013 1 Spanning and linear independence I will outline a slightly different approach to the material in Chapter 2 of Axler

### MORE ON CONTINUOUS FUNCTIONS AND SETS

Chapter 6 MORE ON CONTINUOUS FUNCTIONS AND SETS This chapter can be considered enrichment material containing also several more advanced topics and may be skipped in its entirety. You can proceed directly

### Standard forms for writing numbers

Standard forms for writing numbers In order to relate the abstract mathematical descriptions of familiar number systems to the everyday descriptions of numbers by decimal expansions and similar means,

### 2. Two binary operations (addition, denoted + and multiplication, denoted

Chapter 2 The Structure of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference between

### Lattices and Orders in Isabelle/HOL

Lattices and Orders in Isabelle/HOL Markus Wenzel TU München October 8, 2017 Abstract We consider abstract structures of orders and lattices. Many fundamental concepts of lattice theory are developed,

### Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 5 December 2010 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with

### Set Theory. CSE 215, Foundations of Computer Science Stony Brook University

Set Theory CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Set theory Abstract set theory is one of the foundations of mathematical thought Most mathematical

### CHAPTER 1. MATHEMATICAL LOGIC 1.1 Fundamentals of Mathematical Logic

CHAPER 1 MAHEMAICAL LOGIC 1.1 undamentals of Mathematical Logic Logic is commonly known as the science of reasoning. Some of the reasons to study logic are the following: At the hardware level the design

### Some Background Material

Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important

### 32 Divisibility Theory in Integral Domains

3 Divisibility Theory in Integral Domains As we have already mentioned, the ring of integers is the prototype of integral domains. There is a divisibility relation on * : an integer b is said to be divisible

### Sets. Alice E. Fischer. CSCI 1166 Discrete Mathematics for Computing Spring, Outline Sets An Algebra on Sets Summary

An Algebra on Alice E. Fischer CSCI 1166 Discrete Mathematics for Computing Spring, 2018 Alice E. Fischer... 1/37 An Algebra on 1 Definitions and Notation Venn Diagrams 2 An Algebra on 3 Alice E. Fischer...

### SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS

SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, III. THE CASE OF TOTALLY ORDERED SETS MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a partially ordered set P, let Co(P) denote the lattice of all order-convex

### CSE 1400 Applied Discrete Mathematics Proofs

CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4

### INFINITY: CARDINAL NUMBERS

INFINITY: CARDINAL NUMBERS BJORN POONEN 1 Some terminology of set theory N := {0, 1, 2, 3, } Z := {, 2, 1, 0, 1, 2, } Q := the set of rational numbers R := the set of real numbers C := the set of complex

### Supplementary Material for MTH 299 Online Edition

Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think

### 2. The Concept of Convergence: Ultrafilters and Nets

2. The Concept of Convergence: Ultrafilters and Nets NOTE: AS OF 2008, SOME OF THIS STUFF IS A BIT OUT- DATED AND HAS A FEW TYPOS. I WILL REVISE THIS MATE- RIAL SOMETIME. In this lecture we discuss two

### Semantics of intuitionistic propositional logic

Semantics of intuitionistic propositional logic Erik Palmgren Department of Mathematics, Uppsala University Lecture Notes for Applied Logic, Fall 2009 1 Introduction Intuitionistic logic is a weakening

### Lecture 2: Syntax. January 24, 2018

Lecture 2: Syntax January 24, 2018 We now review the basic definitions of first-order logic in more detail. Recall that a language consists of a collection of symbols {P i }, each of which has some specified

### Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

### arxiv:math/ v1 [math.gm] 21 Jan 2005

arxiv:math/0501341v1 [math.gm] 21 Jan 2005 SUBLATTICES OF LATTICES OF ORDER-CONVEX SETS, I. THE MAIN REPRESENTATION THEOREM MARINA SEMENOVA AND FRIEDRICH WEHRUNG Abstract. For a partially ordered set P,

### 3 COUNTABILITY AND CONNECTEDNESS AXIOMS

3 COUNTABILITY AND CONNECTEDNESS AXIOMS Definition 3.1 Let X be a topological space. A subset D of X is dense in X iff D = X. X is separable iff it contains a countable dense subset. X satisfies the first

### The Real Number System

MATH 337 The Real Number System Sets of Numbers Dr. Neal, WKU A set S is a well-defined collection of objects, with well-defined meaning that there is a specific description from which we can tell precisely

### Sets are one of the basic building blocks for the types of objects considered in discrete mathematics.

Section 2.1 Introduction Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Important for counting. Programming languages have set operations. Set theory

### 1. To be a grandfather. Objects of our consideration are people; a person a is associated with a person b if a is a grandfather of b.

20 [161016-1020 ] 3.3 Binary relations In mathematics, as in everyday situations, we often speak about a relationship between objects, which means an idea of two objects being related or associated one

### Introduction to Metalogic

Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)

### Logic Synthesis and Verification

Logic Synthesis and Verification Boolean Algebra Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2014 1 2 Boolean Algebra Reading F. M. Brown. Boolean Reasoning:

### Abstract Measure Theory

2 Abstract Measure Theory Lebesgue measure is one of the premier examples of a measure on R d, but it is not the only measure and certainly not the only important measure on R d. Further, R d is not the

### 2.2 Lowenheim-Skolem-Tarski theorems

Logic SEP: Day 1 July 15, 2013 1 Some references Syllabus: http://www.math.wisc.edu/graduate/guide-qe Previous years qualifying exams: http://www.math.wisc.edu/ miller/old/qual/index.html Miller s Moore

### A Semester Course in Basic Abstract Algebra

A Semester Course in Basic Abstract Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved December 29, 2011 1 PREFACE This book is an introduction to abstract algebra course for undergraduates

### Contribution of Problems

Exam topics 1. Basic structures: sets, lists, functions (a) Sets { }: write all elements, or define by condition (b) Set operations: A B, A B, A\B, A c (c) Lists ( ): Cartesian product A B (d) Functions

### Three Classes of Orthomodular Lattices 3

International Journal of Theoretical Physics, Vol. 45, No. 2, February 2006 ( C 2006) DOI: 10.1007/s10773-005-9022-y Three Classes of Orthomodular Lattices 3 Richard J. Greechie 1 and Bruce J. Legan 2

### 0.2 Vector spaces. J.A.Beachy 1

J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a

### Maximal perpendicularity in certain Abelian groups

Acta Univ. Sapientiae, Mathematica, 9, 1 (2017) 235 247 DOI: 10.1515/ausm-2017-0016 Maximal perpendicularity in certain Abelian groups Mika Mattila Department of Mathematics, Tampere University of Technology,

### CONSTRUCTION OF THE REAL NUMBERS.

CONSTRUCTION OF THE REAL NUMBERS. IAN KIMING 1. Motivation. It will not come as a big surprise to anyone when I say that we need the real numbers in mathematics. More to the point, we need to be able to

### Math 535: Topology Homework 1. Mueen Nawaz

Math 535: Topology Homework 1 Mueen Nawaz Mueen Nawaz Math 535 Topology Homework 1 Problem 1 Problem 1 Find all topologies on the set X = {0, 1, 2}. In the list below, a, b, c X and it is assumed that

### (D) Introduction to order types and ordinals

(D) Introduction to order types and ordinals Linear orders are one of the mathematical tools that are used all over the place. Well-ordered sets are a special kind of linear order. At first sight well-orders

### Stanford Encyclopedia of Philosophy

Stanford Encyclopedia of Philosophy The Mathematics of Boolean Algebra First published Fri Jul 5, 2002; substantive revision Mon Jul 14, 2014 Boolean algebra is the algebra of two-valued logic with only

### Chapter 4. Basic Set Theory. 4.1 The Language of Set Theory

Chapter 4 Basic Set Theory There are two good reasons for studying set theory. First, it s a indispensable tool for both logic and mathematics, and even for other fields including computer science, linguistics,

### Handout #6 INTRODUCTION TO ALGEBRAIC STRUCTURES: Prof. Moseley AN ALGEBRAIC FIELD

Handout #6 INTRODUCTION TO ALGEBRAIC STRUCTURES: Prof. Moseley Chap. 2 AN ALGEBRAIC FIELD To introduce the notion of an abstract algebraic structure we consider (algebraic) fields. (These should not to

### 1.3. The Completeness Axiom.

13 The Completeness Axiom 1 13 The Completeness Axiom Note In this section we give the final Axiom in the definition of the real numbers, R So far, the 8 axioms we have yield an ordered field We have seen

### Outline. We will now investigate the structure of this important set.

The Reals Outline As we have seen, the set of real numbers, R, has cardinality c. This doesn't tell us very much about the reals, since there are many sets with this cardinality and cardinality doesn't

### Indeed, if we want m to be compatible with taking limits, it should be countably additive, meaning that ( )

Lebesgue Measure The idea of the Lebesgue integral is to first define a measure on subsets of R. That is, we wish to assign a number m(s to each subset S of R, representing the total length that S takes

### Lebesgue Measure. Dung Le 1

Lebesgue Measure Dung Le 1 1 Introduction How do we measure the size of a set in IR? Let s start with the simplest ones: intervals. Obviously, the natural candidate for a measure of an interval is its

### MATH 13 SAMPLE FINAL EXAM SOLUTIONS

MATH 13 SAMPLE FINAL EXAM SOLUTIONS WINTER 2014 Problem 1 (15 points). For each statement below, circle T or F according to whether the statement is true or false. You do NOT need to justify your answers.

### MA651 Topology. Lecture 10. Metric Spaces.

MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### 2 so Q[ 2] is closed under both additive and multiplicative inverses. a 2 2b 2 + b

. FINITE-DIMENSIONAL VECTOR SPACES.. Fields By now you ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices,

### Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

### REVIEW OF ESSENTIAL MATH 346 TOPICS

REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations

### This chapter contains a very bare summary of some basic facts from topology.

Chapter 2 Topological Spaces This chapter contains a very bare summary of some basic facts from topology. 2.1 Definition of Topology A topology O on a set X is a collection of subsets of X satisfying the

### Boolean Algebra and Digital Logic

All modern digital computers are dependent on circuits that implement Boolean functions. We shall discuss two classes of such circuits: Combinational and Sequential. The difference between the two types

### Sets, Logic and Categories Solutions to Exercises: Chapter 2

Sets, Logic and Categories Solutions to Exercises: Chapter 2 2.1 Prove that the ordered sum and lexicographic product of totally ordered (resp., well-ordered) sets is totally ordered (resp., well-ordered).

### CHAPTER 7. Connectedness

CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### Propositional Logics and their Algebraic Equivalents

Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic

### Maths 212: Homework Solutions

Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

### General Notation. Exercises and Problems

Exercises and Problems The text contains both Exercises and Problems. The exercises are incorporated into the development of the theory in each section. Additional Problems appear at the end of most sections.

### The Integers. Math 3040: Spring Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers Multiplying integers 12

Math 3040: Spring 2011 The Integers Contents 1. The Basic Construction 1 2. Adding integers 4 3. Ordering integers 11 4. Multiplying integers 12 Before we begin the mathematics of this section, it is worth

### Löwenheim-Skolem Theorems, Countable Approximations, and L ω. David W. Kueker (Lecture Notes, Fall 2007)

Löwenheim-Skolem Theorems, Countable Approximations, and L ω 0. Introduction David W. Kueker (Lecture Notes, Fall 2007) In its simplest form the Löwenheim-Skolem Theorem for L ω1 ω states that if σ L ω1

### August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei. 1.

August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei 1. Vector spaces 1.1. Notations. x S denotes the fact that the element x

### Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009

Discrete Mathematics: Lectures 6 and 7 Sets, Relations, Functions and Counting Instructor: Arijit Bishnu Date: August 4 and 6, 2009 Our main goal is here is to do counting using functions. For that, we

### Economics 204 Fall 2011 Problem Set 1 Suggested Solutions

Economics 204 Fall 2011 Problem Set 1 Suggested Solutions 1. Suppose k is a positive integer. Use induction to prove the following two statements. (a) For all n N 0, the inequality (k 2 + n)! k 2n holds.

### AN INTRODUCTION TO BOOLEAN ALGEBRAS

California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 12-2016 AN INTRODUCTION TO BOOLEAN ALGEBRAS Amy Schardijn California

### ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS

ADVANCED CALCULUS - MTH433 LECTURE 4 - FINITE AND INFINITE SETS 1. Cardinal number of a set The cardinal number (or simply cardinal) of a set is a generalization of the concept of the number of elements

### Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017

Advanced Calculus: MATH 410 Real Numbers Professor David Levermore 1 November 2017 1. Real Number System 1.1. Introduction. Numbers are at the heart of mathematics. By now you must be fairly familiar with

### CHAPTER 1 INTRODUCTION TO BRT

CHAPTER 1 INTRODUCTION TO BRT 1.1. General Formulation. 1.2. Some BRT Settings. 1.3. Complementation Theorems. 1.4. Thin Set Theorems. 1.1. General Formulation. Before presenting the precise formulation

### CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998

CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic

### CHAPTER 5. The Topology of R. 1. Open and Closed Sets

CHAPTER 5 The Topology of R 1. Open and Closed Sets DEFINITION 5.1. A set G Ω R is open if for every x 2 G there is an " > 0 such that (x ", x + ") Ω G. A set F Ω R is closed if F c is open. The idea is

### REAL AND COMPLEX ANALYSIS

REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any

### NOTES (1) FOR MATH 375, FALL 2012

NOTES 1) FOR MATH 375, FALL 2012 1 Vector Spaces 11 Axioms Linear algebra grows out of the problem of solving simultaneous systems of linear equations such as 3x + 2y = 5, 111) x 3y = 9, or 2x + 3y z =

### We are going to discuss what it means for a sequence to converge in three stages: First, we define what it means for a sequence to converge to zero

Chapter Limits of Sequences Calculus Student: lim s n = 0 means the s n are getting closer and closer to zero but never gets there. Instructor: ARGHHHHH! Exercise. Think of a better response for the instructor.

### However another possibility is

19. Special Domains Let R be an integral domain. Recall that an element a 0, of R is said to be prime, if the corresponding principal ideal p is prime and a is not a unit. Definition 19.1. Let a and b

### Filters in Analysis and Topology

Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

### Metric Space Topology (Spring 2016) Selected Homework Solutions. HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y)

Metric Space Topology (Spring 2016) Selected Homework Solutions HW1 Q1.2. Suppose that d is a metric on a set X. Prove that the inequality d(x, y) d(z, w) d(x, z) + d(y, w) holds for all w, x, y, z X.

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. College Algebra for STEM

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics College Algebra for STEM Marcel B. Finan c All Rights Reserved 2015 Edition To my children Amin & Nadia Preface From

### MA651 Topology. Lecture 9. Compactness 2.

MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology

### Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

### HW 4 SOLUTIONS. , x + x x 1 ) 2

HW 4 SOLUTIONS The Way of Analysis p. 98: 1.) Suppose that A is open. Show that A minus a finite set is still open. This follows by induction as long as A minus one point x is still open. To see that A

### 1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set