DISCRETE MATH: LECTURE 18 DR. DANIEL FREEMAN 1. Chapter 6.1 again! Definition (ordered n-tuple). Let n be a positive integer and let x1; x2; :::; xn be n ele- ments. (x1; x2) is called an ordered pair,(x1; x2; x3) is called an ordered triple, and (x1; x2; :::; xn) is called an ordered n-tuple. Two n-tuples (x1; x2; :::; xn) and (y1; y2; :::; yn) are equal if and only if x1 = y1, x2 = y2, ..., xn = yn. That is: (x1; x2; :::; xn) = (y1; y2; :::; yn) , x1 = y1; x2 = y2; :::; xn = yn Examples: • Is (1; 2; 3) = (1; 2; 3; 4)? • Is f1; 2; 3g = (1; 2; 3)? • Is (1; 2; 3) = (1; 3; 2)? 1 2 DR. DANIEL FREEMAN 1.1. Cartesian Products. Definition. Given sets A1;A2;A3; ::; An, the Cartesian Product of A1;A2; :::; An de- noted A1 × A2 × ::: × An is the set of all ordered n-tuples (a1; a2; :::; an) where a1 2 A1; a2 2 A2; :::; an 2 An. Symbolically, that is: A1 × A2 × ::: × An = f(a1; a2; :::; an) k a1 2 A1; a2 2 A2; :::; an 2 Ang: In particular, A1 × A2 = f(a1; a2) k a1 2 A1; a2 2 A2g: Example: A = x; y, B = 1; 2; 3, and C = a; b • A × B = • (A × B) × C = • A × B × C = DISCRETE MATH: LECTURE 18 3 2. 7.1 Functions Definition. A function f from a set X to a set Y , denoted f : X ! Y , is a relation with domain X and co-domain Y that satisfies the two properties: (1) every element in X is related to an element in Y . (2) no element in X is related to more than one element in Y . For each x 2 X, we denote f(x) to be the element related to x by f. We call the set of values in Y which f maps an element in X to the range of f, or the image of X under f. That is, range of f = fy 2 Y j 9x 2 X; such that f(x) = yg: Given an element y 2 Y , we call all the points which get mapped to y the preimage of y or inverse image of y. That is, preimage of y = fx 2 X j f(x) = yg: Example: Arrow diagrams for X = f0; 1; 2; 3; 4; 5g, Y = f0; 1; 2g • f : X ! Y , f(x) = x mod 2 • g : X ! Y , f(x) = x mod 3 • h : X ! Y , f(x) = 2x mod 6 Two functions f : X ! Y and g : X ! Y are equal if 8x 2 X, f(x) = g(x). 4 DR. DANIEL FREEMAN 2.1. Examples of functions. (1) The identity function: IX : X ! IX is the function: IX (x) = x 8x 2 X: (2) Sequences: Let A be a set, and let a1; a2; a3; ::: be a sequence in A define a function f : N ! A by f(n) = an 8n 2 N: Then f(1); f(2); f(3); ::: is the sequence a1; a2; a3; ::: Let g : N ! A be a function. Then g(1); g(2); g(3); ::: is a sequence in A. Thus sequences are equivalent to functions with domain N. (3) Cartesian product: Let f : X ! Y be a function. Then f(x; f(x)) j x 2 Xg ⊆ X × Y: Let A ⊆ X × Y . What properties must A posses for there to exist a function f : X ! Y such that A = f(x; f(x)) j x 2 Xg ? DISCRETE MATH: LECTURE 18 5 Definition. If f : X ! Y is a function and A ⊆ X and C ⊆ Y , then we denote • f(A) = fy 2 Y j y = f(x) for some x 2 Ag • f −1(C) = fx 2 X j f(x) 2 Cg f(A) is called the image of A, and f −1(C) is called the inverse image of C. Example: Let f : X ! Y be a function and A; B ⊆ X. Prove that: f(A \ B) ⊂ f(A) \ f(B) Give an example of a function f : X ! Y and sets A; B ⊆ X such that f(A \ B) 6= f(A) \ f(B): 6 DR. DANIEL FREEMAN In class work: (1) Let f : X ! Y be a function and A; B ⊆ X. Prove that: f(A [ B) = f(A) [ f(B) DISCRETE MATH: LECTURE 18 7 (2) Let f : X ! Y be a function and A ⊆ X. Prove that: A ⊆ f −1(f(A)): (3) Give an example of a function f : X ! Y and a set A ⊆ X such that: A 6= f −1(f(A)) 8 DR. DANIEL FREEMAN (4) Let f : X ! Y be a function and C; D ⊆ Y . Prove that: f −1(C \ D) = f −1(C) \ f −1(D).