Foundations Revision Notes

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1 oundations Revision Notes hese notes are designed as an aid not a substitute for revision. A lot of proofs have not been included because you should have them in your notes, should you need them. Also, the course has changed somewhat since when this guide was written, so you will need to exercise some caution as not all the material here may be relevant anymore. Please send any comments or suggestions to Mathsoc. If anyone is willing to lend mathsoc their notes from this year then I can update the guide for next year. Happy revising and good luck in your exams! Chapter 0: Integers Z is the set of integers N is the set of natural numbers (i.e. positive integers, not including 0). Definition: We say b divides a (or a is divisible by b) if and only if c Z s.t. a = bc. We write b a to denote this. Definition: a Z,b N, q, r Z, 0 r < b such that a = bq +r. We call q the quotient and r the remainder. Definition: or a, b N, the highest common factor hcf {a, b} = max{d N d a and d b}. or a, b N, the lowest common multiple lcm {a, b} = min{d N a d and b d}. a, b N hcf{a, b} lcm{a, b} = ab. heorem: Euclid s Algorithm Let a, b be integers, with b > 0. o find hcf{a, b}, repeat division with remainder as follows: a = q 0 b + r 1 (r 1 < b) b = q 1 r 1 + r 2 (r 2 < r 1 ) r 1 = q 2 r 2 + r 3 (r 3 < r 2 ).. r t 2 = q t 1 r t 1 + r t (r t < r t 1 ) r t 1 = q t r t + 0 his process always terminates in this way, so that r t, the last non-zero remainder equals hcf{a, b} and x, y Z so that r t = xa + yb. heorem: If h = hcf {a, b} and d = xa + yb for somex, y Z then h d. Definition: p N is a prime number is it has exactly two factors, 1 and p i.e. {n N m N if m n then m = n or m = 1}. 1

2 Note: 1 is not a prime number. Lemma: If p is prime, a, b N with p ab then either p a or p b or both. heorem: here are an infinite number of primes. Proof by contradiction. heorem: undamental heorem of Arithmetic Every natural number can be expressed as the product of prime numbers, and this expression is unique up to the order of the factors. Chapter 1: Sets he set is the one undefined concept in mathematics. A set is determined entirely by its elements. he elements of a set are unordered and no element can occur twice in the same set. Any object can belong to a set - elements are not restricted to numbers or other mathematical objects. It must be possible to decide whether or not any object belongs to a given set. Definition: A subset B of a set A is defined as a set with the property that b B b A i.e. every element of B is an element of A. If B is a subset of A we write B A. N Z Q R C. Definition: he empty set, denoted is the set containing nothing. or all sets A, A and A A are trivial subsets. A=B if and only if A B and B A. Set Operations: intersection x A B if x A and x B. union x A B if x A or x B or both. \ set difference x A\B if x A but x B. Definition: Universe of discourse We can define a set U to contain all elements currently under consideration. We can then define a new operation A c = U\A, x A c if x U but x A. We can also say that A A c = U and A A c =. heorem: Russell s Paradox Let S = {Ω Ω Ω} i.e. the set of all sets which do not contain themselves. Is S S? By definition of S, S S S S and S S S S. his is a contradiction so S cannot be a valid set. Moral - we have to be careful about what we allow to be a set i.e a set of all sets does not exist. 2

3 De Morgan s Laws (A B) c = A c B c (A B) c = A c B c Distributive Laws A (B C) = (A B) (A C) A (B C) = (A B) (A C) Definition: Ordered pairs (x, y) = (x, y ) if and only if x = x and y = y. Definition: he cartesian product A B is the set of all ordered pairs (a, b) such that a A and b B. Definition: or a finite set A, the number of elements in A is called the cardinality and denoted A. A B = A B A B A B B A A (B C) = (A B) (A C) Definition: he power set of a set A, is the set of all subsets of A, and is denoted P (A). Elements of P (A) are always sets. e.g. A = {x, y, z} : P (A) = {, {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}}. In general if X = n then P (X) = 2 n. Definition: A partition of X is a collection of subsets A i X s.t. A i = X and A i A j = if i j. Chapter 2: unctions Definition: Given two sets D and, a function f : D is a subset of D with the following properties: or each d D t such that (d, t) f i.e. everything is mapped from. If (d, t 1 ) and (d, t 2 ) both belong to f, then t 1 = t 2 i.e. everything is mapped by f to one and only one element of. D is called the domain and is called the target or codomain. We write f(a) = b for (a, b) f. Definition: he image of a funtion f : D is {f(d) d D} and is denoted Im(f). Definition: A map f is injective is f(a) = f(b) a = b i.e. everything that is mapped to is only mapped to once. Definition: A map f : D is surjective is t everything is mapped to. d D such that f(d) = t i.e. 3

4 Definition: A map f is bijective if it is injective and surjective i.e. there is a one-to-one correspondence between the domain and target. Equality of maps f = g if and only if f, g : D and f(x) = g(x) x D. Note: for two maps to be equal they must have the same domain and target. In general for finite sets D and, f : D D surjective D injective D = bijective Composition of Maps If f : A B and g : B C are injective, then (g f)(x) = g(f(x)) x A. Note: g f means apply f first and then g. If f : A B and g : B C are injective, then (g f) : A C is injective. If f : A B and g : B C are surjective, then (g f) : A C is surjective. If (g f) is surjective, then f is surjective. If (g f) is injective, then g is injective. f : D is surjective if and only if Im(f) =. If f : D is injective and D is finite, then D = Im(f). he best way to understand these is to draw simple diagrams, but remember that a picture generally isnt a proof. Lemma: Suppose f : A B, g : B C, h : C D are all well defined functions. hen h (g f)(x) = ((h g) f)(x) i.e. composition of functions is associative. Definition: If f : A B is a bijective map, we can define f 1 : B A such that f 1 (b) = {a A f(a) = b}. Definition: or a set A, the function id A : A A is the identity function such that id A (a) = a a A. heorem: If f : A B is bijective then a function g : B A s.t. g f = id A and f g = id B. 1 A 1 B because they have different domains heorem: Schroder Bernstein If for 2 sets A and B injections f : A B and g : B A then a bijection h : A B. his is true for finite and infinite sets. heorem: Z = Q Note: there is not a bijection between Z and R. 4

5 heorem: Cantor s Diagonal Argument N = R. Definition: N = ℵ 0 aleph zero. If a set A has cardinality ℵ 0 then we say it is countably infinite. If A is countably infinite or finite, we say it is countable. Definition: ℵ 1 aleph one, R = P(N) = 2 ℵ 0. Hilbert s Hotel: i) ℵ 0 + r = ℵ 0 r > 0 ii) rℵ 0 = ℵ 0 r > 0 Chapter 3: Logic Definition: A statement which is always true or false is called a proposition. Definition: Given a proposition P, we define P to be the proposition P is not the case. Definition: A predicate is a statement which contains some unbound variables which determine if it is true or false. e.g. P (x) : x + 2 = 4. A predicate evaluated at a point in its domain becomes a proposition. e.g. P (2) : = 4 true. Definition: Compound statements can be formed from simple statements using logical connectives. We define three logical connectives using truth tables: P Q means P or Q. P Q means P and Q. If P Q then we say P implies Q; i.e. P is sufficient for Q, and Q is necessary for P. 5

6 P P P Q P Q P Q P Q P Q P Q Definition: wo compound statements are logically equivalent if they have the same truth table. If A and B are logically equivalent we can write A B, ora B. De Morgan s Laws (P Q) ( P ) ( Q) (P Q) ( P ) ( Q) Note: ( P ) P ; ( P ) Q P Q Definition: A compund statement is a tautology if it is always true irrespective of the atomic statements. P P ( P ) Definition: A compound statement is a contradiction if it is false irrespective of the atomic statements. P P ( P ) Definition: Negating statements - swap the s and the and negate the statement. e.g. ( xp (x)) x P (x). 6

7 Chapter 4: Methods of Proof Contrapositive: In order to prove P Q we can prove the logically equivalent statement Q P. Contradiction: In order to prove P, we can prove P C where C is a contradiction. Induction: A proposition is true n N if: P (1) is true P (k) P (k + 1) for any k N. Chapter 5: Relations Definition: A relation on a set X is a subset of X X, i.e. a set of ordered pairs of elements of X. We say that for any a, b X, a is related to be (written a b) if and only if (a, b) Note: Since we use ordered pairs a b and b a are NO necessarily equivalent statements. Definition: A relation on a set X is called a partial order if it satisfies the following properties: Reflexive: x x, x X Anti-symmetric: x y and y x then x = y ransitive: x y and y z then x z Definition: A relation on a set X is a total order if it is transitive and satisfies the additional property: richotomy: x, y X either x y or y x (or both). Definition: A relation on a set X is called an equivalence relation if it satisfies the following three properties: Reflexive: x x, x X Symmetric: if x y then y x ransitive: if x y and y z then x z Definition: When is an equivalence relation on S, then the sets E x = {y S y x} are called equivalence classes, also written [x]. hen x, y S, [x] = [y] or [x] [y] = and S is partitioned by its equivalence classes. heorem: If is an equivalence relation in S then P = {E x x S} is a partition of S. Conversely if we have a partition of a set S then this partition defines an equivalence relation. 7

8 Chapter 6: Permutations and Groups Definition: A permutation of a set X is a bijection from X to itself. We define Sym(X) = {ρ ρ : X X is a bijection }. It is the set of symmetries (permutations) of X. Properties of (Sym(X), ) i) Closure: the set is closed under composition i.e. if π, ρ Sym(X) then π ρ Sym(X). ii) Identity: there is an element e Sym(X) such that e ρ = ρ e = ρ ρ Sym(X) e = id x the identity function id x (x) = x. iii) Inverse: given any ρ Sym(X) unique ρ 1 such that ρ ρ 1 = ρ 1 ρ = e. iv) Associative: ρ, π, σ Sym(X) then (ρ π) σ = ρ (π σ). Definition: S n = Sym({1, 2,..n}). If X = n then Sym(X) = n! Notation: 2-row notation 1 2 n σ = σ(1) σ(2) σ(n) Example of composition: If ρ = and σ = then ρ σ = and σ ρ = i.e. (ρ σ)(1) = ρ(σ(1)) = ρ(4) = 4 Note: In general, S n does not commute i.e. ρ, σ S n ρ σ σ ρ Notation: Disjoint Cycle Notation An alternative way of representing a permutation is as a number of disjoint cycles. or a permutation σ of {1, 2,...n} the first cycle is (1, σ(1), σ(σ(1)), ) stopping at the last element before we reach 1 again, so if σ n (1) = 1 then we stop at σ n 1 (1). If any of the 1,...n are not included in this cycle, we start from the first element not included and form another cycle from this element in the same way. We continue until all the elements are included in a cycle. or example: = (152)(36)(4) hen each element is mapped to the next element in its cycle. Read the brackets right to left, but read inside the brackets left to right. Note: often singleton elements - like 4 in the above example - are not included, as they have no effect. 8

9 Example of composition Let σ = (15723)(46) and ρ = (172)(3546) so σ ρ = (15723)(46)(172)(3546) = (1256)(37)(4) inding inverse permutations 2-row notation Given a permutation π we wish to find π 1. If we write π in 2-row notation then it is relatively easy to find the inverse, simply flip the permutation upside down, and then rearrange the columns so that the top row is in the correct order. or example Disjoint cycle notation o invert a permutation written in disjoint cycle notation, simply reverse the order of the elements in each bracket, and then remember that the lowest numbered element must be first in each bracket. or example: (152)(36) (251)(63) (152)(36) Definition: A permutation of the form (ab) is a transposition, and every cycle can be written as a non-unique product of transpositions. Example: (16345) = (16)(63)(34)(45) = (61)(36)(43)(54) Definition: A permutation is called odd if it can be written as an odd number of transpositions. It is called even if it can be written as an even number of transpositions. e.g. (16345) is even. Definition: he order of a permutation ρ is the smallest k such that ρ k = e (identity). In fact, order ρ = lcm{ lengths of cycles in ρ} Definition: A binary operation on A is a map from A A to A, which maps each ordered pair of elements of A, (x, y) to an element x y A. e.g. addition and multiplication on R are binary operations. Definition: A set G with binary operation is called a group of it satisfies: Closure: g h G g, h G Associativity: (g h) k = g (h k) g, h, k G Identity: e G such that g e = e g = g g G Inverse: g G h G such that g h = h g = e Note: Closure says that is a binary operation, it is included in the properties to remind you to check for it. Definition: If we have g h = h g g, h G then (G, ) is commutative or abelian. 9

10 We can draw Cayley ables to see whether something is a group or not. e.g. (Z 6, + 6 ) i.e. Z 6 = {0, 1, 2, 3, 4, 5}; + 6 = addition modulo identity: yes - x + 0 = 0 inverse: yes - every row has a 0. associative: yes - addition is associative closure: yes - table only contains {0, 1, 2, 3, 4, 5} heorem: Latin Square Property In each row and column of a Cayley able for a group (G, ) the elements of G appear exactly once (each column and row is a permutation of G). heorem: If (G, ) is a group then there is only one identity e. heorem: If (G, )is a group and g G then g has only one inverse. heorem: Cancellation Law If (G, ) is a group with g, a, b G and ga = gb then a = b. Definition: Let (G, ) be a group. A subgroup of G is a non-empty subset H G such that (H, ) is itself a group. o prove that a subset is a subgroup, only 2 properties need to be checked: h k H h, k H h H h 1 H. Associativity is preserved because the binary operation is the same as in G, and it should be clear that the identity element of H must be the same one as in G. Definition: If (G, ) is a group with an element g G such that g = {g, g 2, g 3,...} then G is cyclic and G is generated by g, G =< g >. his is finite because g k = e for some k N. Lagrange s heorem If G is a finite group and H is a subgroup of G then G is a multiple of H i.e. H divides G. Note: it doesnt say that if n G then there exists a subgroup H such that H = n. 10