Chapter 1 : The language of mathematics.


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1 MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or : P Q P or Q T T T T F T F T T F F F Truth table for the connective and : P Q P and Q T T T T F F F T F F F F Truth table for the connective not : P T F not P F T Chapter 2 : Implications. Truth table for the implication : P Q P Q T T T T F F F T T F F T Chapter 3 : Proofs. Trichotomy law : For real numbers a and b, one and only one of the three possibilities a < b, a = b, a > b is true. 1
2 2 Addition law : For real numbers a, b and c, a < b a + c < b + c. Multiplication law : For real numbers a, b and c, a < b ac < bc if c > 0; a < b ac > bc if c < 0. Transitive law : For real numbers a, b and c, a < b and b < c a < c. Chapter 4 : Proof by contradiction. Consists of proving the statement (not P ) Q where Q is a false statement. Proving an implication P Q by contradiction : assume P is true and Q is false, and obtain a contradiction. Proof by contrapositive : based on the fact that the statements P Q and its contrapositive (not Q) (not P ) are logically equivalent. Proving or statements : based on the fact that the statements P or Q and (not P ) Q are logically equivalent. Proposition. Given nonzero real numbers a and b, we have ab > 0 if and only if a and b have the same sign, and ab < 0 if and only if a and b have opposite signs. Chapter 5 : The Induction Principle. Axiom (The induction principle). Suppose that P (n) is a statement involving a general positive integer n. Then P (n) is true for all positive integers 1, 2, 3,... if (i) P (1) is true, (ii) P (k) P (k + 1) for all positive integers k. Chapter 6 : The language of set theory. Definition. A set is any welldefined collection of objects. Examples : Z is the set of all integers.
3 Z is the set of all nonnegative integers 0, 1, 2, 3,... N is the set of all positive integers 1, 2, 3,... Q is the set of all rational numbers (fractions). R is the set of all real numbers. Definition. The objects in a set are called elements, members or points. We write x E to denote the fact that the object x is an element of the set E. Specifying a set : Listing the elements, example : A = {1, 3, π, 5}. The order is not important and there is no repetition. Conditional definition, example : A = {n N : 1 < n < 8}. Constructive definition, example : A = {n 2 +1 : n N} = {2, 5, 10, 17,... }. Definition. Two sets A and B are equal, written A = B, if they have precisely the same elements, i.e. x A x B. Definition. The empty set is the unique set which has no elements at all. It is denoted by. Definition. Given sets A and B, we say that A is a subset of B, written A B, if every element of A is an element of B, i.e. x A x B. In this case, if in addition A is not equal to B (so that B contains an element which is not an element of A), then we say that A is a proper subset of B and write A B. Thus A = B if and only if A B and B A. Operations on sets : Intersection : A B = {x : x A and x B}. Union : A B = {x : x A or x B}. Difference : A \ B = {x : x A and x / B}. Definition. Two sets A and B are said to be disjoint if A B =. Definition. The power set of a set X, denoted by P(X), is the set of all subsets of X. Thus A P(X) if and only if A X. Definition. Once we have fixed a universal set U, we can define the complement of any A P(U), denoted by A c, by A c = U \ A. Theorem. Let A, B, C be subsets of some universal set U. Then (i) Associativity : A (B C) = (A B) C, A (B C) = (A B) C; (ii) Commutativity : A B = B A, A B = B A; (iii) Distributivity : A (B C) = (A B) (A C), A (B C) = (A B) (A C); (iv) De Morgan laws : (A B) c = A c B c, (A B) c = A c B c ; 3
4 4 (v) Complementation : A A c = U, A A c = ; (vi) Double complements : (A c ) c = A. Chapter 7 : Quantifiers. Universal statement : a A, P (a) means that for all elements a in the set A, the proposition P (a) is true. Example : n N, n > 0. Existential statement : a A, P (a) means that there exists an element a in the set A for which the proposition P (a) is true. Example : x R, x 2 = π. Proving and disproving statements involving quantifiers : Proving statements of the form a A, P (a) : it suffices to prove the implication a A P (a). Proving statements of the form a A, P (a) : it suffices to exhibit a particular element a A for which P (a) is true. Disproving statements of the form a A, P (a) : we prove the negation a A, not P (a) by giving a counterexample, i.e. an element in A for which P (a) is false. Disproving statements of the form a A, P (a) : we prove the negation a A, not P (a). Definition. Given sets X and Y, the Cartesian product of X and Y, denoted by X Y, is the set of all ordered pairs (x, y) where x X and y Y X Y = {(x, y) : x X, y Y }. When Y = X, we write X X = X 2. Chapter 8 : Functions. Definition. Let X and Y be sets. A function f : X Y is the assignment of a unique element of Y to each element of X. We denote the element of Y assigned to x X by f(x); it is called the value of f at x or the image of x under f. The set X is called the domain of the function f and the set Y is called the codomain. Definition. The identity function I X : X X is defined by I X (x) = x for all x X. Definition. Two functions f : X Y and g : X Y are equal, written f = g, if they have the same value at each point of the domain X, i.e. f(x) = g(x) for all x X.
5 Definition. Given two functions f : X Y and g : Y Z, the composite of f and g, denoted by g f : X Z, is the function defined by (g f)(x) = g(f(x)) (x X). Proposition. Suppose that f : X Y, g : Y Z and h : Z W are functions. Then (i) (h g) f = h (g f). (ii) f I X = f = I Y f. Definition. Given a function f : X Y, we define its image by Im f = {f(x) : x X}. It is the subset of the codomain Y consisting of all values of f. Definition. Suppose that f : X Y is a function. Then we define the graph of f to be the subset of X Y given by G f = {(x, f(x)) : x X}. 5 Chapter 9 : Injections, surjections and bijections. Definition. Suppose that f : X Y is a function. (i) We say that f is injective if x 1, x 2 X, (f(x 1 ) = f(x 2 ) x 1 = x 2 ); (ii) We say that f is surjective if y Y, x X, y = f(x); (iii) We say that f is bijective if it is both injective and surjective. Definition. For a function f : X Y and an element y Y, we say that an element x X is a preimage of y under f if y = f(x). Thus f is injective if and only if every element of Y has at most one preimage. f is surjective if and only if every element of Y as at least one preimage. f is bijective if and only if every element of Y has exactly one preimage. Definition. A function f : X Y is invertible if there exists a function g : Y X such that y = f(x) x = g(y) ( x X, y Y ). In this case, we say that g is the inverse of f and write g = f 1. The symmetry of the definition shows that in this case g is also invertible and f is the inverse of g.
6 6 Theorem. Let f : X Y be a function. Then f is invertible if and only if it is bijective. Furthermore, if this is the case, then the inverse of f is unique. Proposition. The functions f : X Y and g : Y X are inverses to each other if and only if g f = I X and f g = I Y. Definition. Let f : X Y be a function. (i) The function f : P(X) P(Y ) is defined by f(a) = {f(x) : x A} for A P(X); (ii) The function f 1 : P(Y ) P(X) is defined by f 1 (B) = {x X : f(x) B}. Chapter 10 : Counting. Given n N, we define N n = {1, 2,..., n}. Definition. Let X be a set. If there is a bijection f : N n X, then we say that the cardinality of X is n and write X = n. The cardinality of the empty set is defined to be 0. Proposition. Suppose that f : N m X and g : N n X are bijections. Them m = n. Definition. Let X be a set. If X = n for some nonnegative integer n, then we say that X is finite. Otherwise, we say that X is infinite. Theorem (Addition principle). Suppose that X and Y are disjoint finite sets. Then X Y is finite and X Y = X + Y. Corollary. Suppose that X 1, X 2,..., X n are pairwise disjoint finite sets. Then X 1 X 2 X n is finite and X 1 X 2 X n = X 1 + X X n. Theorem (Multiplication principle). Suppose that X and Y are finite sets. Then X Y is finite and X Y = X Y. Proposition (Inclusionexclusion principle for two sets). Suppose that X and Y are finite sets. Then X Y is finite and X Y = X + Y X Y.
7 Proposition (Inclusionexclusion principle for three sets). Suppose that X, Y and Z are finite sets. Then X Y Z is finite and X Y Z = X + Y + Z X Y X Z Y Z + X Y Z. 7 Chapter 11 : Properties of finite sets. Theorem (Pigeonhole principle). Let X and Y be nonempty finite sets. If there exists an injection X Y, then X Y. Equivalently, if X > Y, then every function f : X Y is not injective. Theorem. Let X and Y be nonempty finite sets. If there exists a surjection X Y, then X Y. Equivalently, if X < Y, then every function f : X Y is not surjective. Proposition. Suppose that X Y where Y is a finite set. Then X is also finite and X Y. Theorem. Let X and Y be two nonempty finite sets and suppose that X = Y. Then a function f : X Y is injective if and only if it is surjective. Definition. Let A R. We say that b is a minimal element of A, written b = min A, if (i) b A; (ii) a A b a. Similarly, we say that c is a maximal element of A, written c = max A, if (i) c A; (ii) a A c a. The maximal element and minimal element, if they exist, are unique. Proposition. Let A be a finite nonempty set of real numbers. Then A has a minimum element and a maximum element. Chapter 12 : Counting functions and subsets. Proposition (Number of functions). Suppose that X and Y are nonempty finite sets with X = m and Y = n. Then Fun(X, Y ) = n m. Proposition (Number of injective functions). Let X and Y be nonempty finite sets with X = m and Y = n. Suppose that m n. Then Inj(X, Y ) = n! (n m)!.
8 8 Definition. Given a set X, a bijection X X is called a permutation of the set X. Proposition (Number of permutations). Suppose that X is a finite nonempty set of cardinality n. Then the number of permutations of X is n!. Proposition (Number of subsets). Suppose that X is a finite nonempty set. Then P(X) = 2 X. Definition. Given a set X and a nonnegative integer r, an rsubset of X is a subset A X of cardinality r. We denote by P r (X) the set of all rsubsets of X : P r (X) = {A X : A = r}. We define the binomial coefficient ( n r) to be the cardinality of Pr (X) when X = n. Proposition. For n and r nonnegative integers, we have (i) ( n r) = 0 if r > n; (ii) ( ( n 0) = 1, n ( 1) = n, n n) = 1; (iii) ( ) ( n r = n n r) for 0 r n. Proposition. n j=0 ( ) n = 2 n. j Proposition (Pascal s law). For positive integers n and r such that 1 r n, we have ( ) ( ) ( ) n n 1 n 1 = +. r r 1 r Theorem. For nonnegative integers n and r such that r n, we have ( ) n n! = r r!(n r)!. Theorem (Binomial theorem). For all real numbers a and b and nonnegative integers n, we have n ( ) n (a + b) n = a n j b j. j j=0 Chapter 13 : Number systems. Theorem (Irrationality of 2). There does not exist a rational number whose square is 2.
9 9 Chapter 14 : Counting infinite sets. Definition. Two sets X and Y are equipotent if there is a bijection X Y. Definition. A set X is said to be denumerable if there is a bijection N X. A set is countable if it is either finite or denumerable. A set is uncountable if it is not countable. If X is denumerable, then we say that its cardinality is ℵ 0 and we write X = ℵ 0. Examples of denumerable sets : Z. Z. the set of even integers. Proposition. Let X and Y be sets and suppose that X is denumerable. Then Y is also denumerable if and only if it is equipotent to X. Proposition. If A and B are denumerable, then so is their union A B and their Cartesian product A B. Proposition. A subset of a denumerable set is countable. Theorem (Denumerability of the rationals). The set of rational numbers Q is denumerable. Theorem (Uncountability of the reals). The set of real numbers R is uncountable. Definition. We say that two (possibly infinite) sets X and Y have the same cardinality, written X = Y, if they are equipotent. If there is an injection X Y, then we write X Y. We write X < Y to mean that X Y but X Y. Theorem. For any set X, we have X < P(X). Theorem (Cantor Schröder Bernstein theorem). Suppose that X and Y are nonempty sets with X > Y. Then any function f : X Y is not injective. Corollary. Let X and Y be nonempty sets. If X Y and Y X, then X = Y. Definition. A real number x is called algebraic if it satisfies a polynomial equation a 0 + a 1 x + + a n x n = 0, where the coefficients a 0, a 1,..., a n are integers. If x is not algebraic, then we say that it is transcendental.
10 10 Examples of algebraic numbers : any rational number. the square root of any integer. Proposition. The set of algebraic numbers is denumerable. Chapter 15 : The division theorem. Theorem. Let a, b be integers with b > 0. Then there are unique integers q and r such that a = qb + r and 0 r < b. The integer q is called the quotient and r is called the remainder. Note that b divides a if and only if r = 0. Chapter 16 : The Euclidean algorithm. Definition. Let a and b be two integers, not both zero. The greatest common divisor of a and b, noted (a, b), is the unique positive integer d such that (i) d divides b and d divides a; (ii) If c divides a and c divides b, then c d. The Euclidean algorithm is a procedure which gives the greatest common divisor of two integers a and b. It is based on the following lemmas. Lemma. If a positive integer b divides a, then (a, b) = b. Lemma. Let a and b be nonzero integers and suppose that a = qb + r for some q, r Z. Then (a, b) = (b, r). Chapter 17 : Consequences of the Euclidean algorithm. Theorem. Let a and b be integers not both zero. Then there exist integers m and n such that ma + nb = (a, b). The integers m, n can be obtained by going backwards in the Euclidean algorithm. Corollary. Let a and b be integers not both zero. divisor of a and b if and only if c divides (a, b). Then c is a common
11 Definition. Two integers a and b not both zero are called coprime if (a, b) = 1, in other words their only common divisors are 1 and 1. Proposition. Let a and b be two integers not both zero. Then a and b are coprime if and only if there exist integers m and n such that ma + nb = 1. Theorem. Let a, b and c be positive integers with a and b coprime. If a divides bc, then a divides c. 11 Chapter 18 : Linear Diophantine equations. Theorem. For positive integers a, b and c, there exist integers m and n such that ma + nb = c if and only if (a, b) divides c. Theorem. Let a, b, c be positive integers and suppose that (a, b) divides c. Let m 0, n 0 be integers such that Then m, n is another solution to if and only if and for some q Z. m 0 a + n 0 b = c. ma + nb = c m = m 0 + n = n 0 b (a, b) q a (a, b) q Chapter 19 : Congruence of integers. Definition. Let m be an integer greater than one. We say that the integers a and b are congruent modulo m if m divides a b. In this case, we write a b mod m. Proposition. (i) Reflexive property : For all integers a, a a mod m; (ii) Symmetric property : If a and b are integers such that a b mod m, then b a mod m; (iii) Transitive property : If a, b and c are integers such that a b mod m and b c mod m, then a c mod m.
12 12 Proposition (Modular arithmetic). Suppose that a 1, a 2, b 1, b 2 are integers such that a 1 a 2 mod m and b 1 b 2 mod m. Then (i) a 1 + b 1 a 2 + b 2 mod m; (ii) a 1 b 1 a 2 b 2 mod m; (iii) a 1 b 1 a 2 b 2 mod m. Definition. The set of integers R m = {0, 1, 2,..., m 1} is called the set of residues modulo m. Proposition. Given an integer a, there is a unique r R m such that a r mod m. Proposition (Division in congruences). We have ab 1 ab 2 mod m b 1 b 2 mod m/(a, m). In particular, if a divides m, then ab 1 ab 2 mod m b 1 b 2 mod m/a and if a and m are coprime, then ab 1 ab 2 mod m b 1 b 2 mod m. Chapter 23 : The sequence of prime numbers. Definition. A positive integer n is said to be prime if n > 1 and the only positive divisors of n are 1 and n. If an integer n > 1 is not prime, then it is said to be composite. Proposition. Every integer greater than 1 can be written as a product of prime numbers. Theorem. Let a, b be positive integers and let p be a prime number. If p divides ab, then p divides a or p divides b. Proposition. If n 2 is a composite number, then it has a prime divisor not exceeding n. Theorem (Fundamental theorem of arithmetic). Every positive integer greater than 1 can be written uniquely as a product of prime numbers with the prime factors in the product written in nondecreasing order. Theorem. There are infinitely many prime numbers.
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