Functions. Definition 1 Let A and B be sets. A relation between A and B is any subset of A B.
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1 Chapter 4
2 Functions Definition 1 Let A and B be sets. A relation between A and B is any subset of A B. Definition 2 Let A and B be sets. A function from A to B is a relation f between A and B such that For every element a A there is b B such that (a, b) f. If (a, b),(a, c) f then b = c.
3 We write f : A B if f is a function from A to B. If f : A B thena is called the domain of f and B is called the co-domain of f. The image of a set S A by a function f : A B is the set f(s) = {f(a) a S}. The pre-image of a set T B by a function f : A B is the set f 1 (T) = {a f(a) T }.
4 Some Examples Polynomial function f : R R given by f(x) = n i=1 a i x i where a i R and n N {0}. f : R R given by f(x) = x if x is rational and f(x) = 0 if x is irrational. f : R Z given by f(x) is the least integer which is greater than or equal to x. (Ceiling function). f : R Z given by f(x) is the largest integer which is smaller than or equal to x. (Floor function).
5 Characteristic function. Let A U, f : U {0,1} given by f(x) = 1 if x A and f(x) = 0 if x / A. f : A P(A) given by f(x) = A {x}. f : P(Z + ) Z + {0} given by f(a) is the smallest integer z with z A if Z and f( ) = 0. f : Z + Z + given by f(x) = f(f(x/2)) if x is even and 1 if x is odd.
6 When are functions f and g equal? Functions f, g are equal if f : A B, g : A B and for every a A, f(a) = g(a), that is the sets f, g A B are such that f = g.
7 Injective, surjective, bijective functions Definition 3 Let f : A B. f is called injective (one-to-one) if for every x, y A if f(x) = f(y) then x = y. f is called surjective (onto) if for every b B there is a A with f(a) = b. f is called a bijection (one-to-one correspondence) if f is both injective and surjective.
8 Injection Surjection Bijection
9 Example 1 Check which of the functions below are injective, which are surjective. f : N N, f(x) = x + 7. f : Z Z, f(x) = x + 7. f : N N, f(x) = x f : Z Z, f(x) = x f : N N, f(x) = x 5.
10 f : N N Z, f((x, y)) = x y. f : Z Z Z, f((x, y)) = x + y + 3. f : Z Z Z, f(x) = (x,3x). f : R R, f(x) = x 2 1 if x 0 and f(x) = x 2 if x < 0. f : R R, f(x) = x 2 if x 1 and f(x) = 1 + x if x < 1.
11 Increasing and decreasing functions Definition 4 Let A, B R and let f : A B. f is called increasing when for every x, y A, if x < y then f(x) < f(y). f is called decreasing when for every x, y A, if x < y then f(x) > f(y).
12 Composition of functions Definition 5 Let f : A B and g : B C. Then g f is a function from A to C defined as follows (g f)(x) = g(f(x)). Example 2 Show that f : A B and g : B C are both surjective then g f : A C is surjective. Show that f : A B and g : B C are both injective then g f : A C is injective.
13 Example 3 Decide if the statements are true or false. Let f : A B and g : B C. If g f is injective then f is injective. If g f is injective then g is injective. If g f is surjective then f is surjective. If g f is surjective then g is surjective.
14 Inverse of a function We say that a function f : B is invertible if there exists a function g : B A such that for all a A, b B f(a) = b if and only if g(b) = a. Theorem 1 Let f : A B. Then f is invertible if and only if it is a bijection. Moreover, if f has an inverse then it is unique. If f : A B then the inverse of f is denoted by f 1. Note that f f 1 = id B, f 1 f = id A.
15 Proof of Theorem 1. Suppose g is an inverse of f. Show that f must be injective and surjective. Suppose f is a bijection. Define g(b) to be equal to a such that f(a) = b. Check that g is a function. Suppose g 1, g 2 are inverses of f. Show that g 1 = g 2.
16 Denumerable sets When two finite sets A, B have same cardinality? When there is a bijection f : A B. Definition 6 Two sets A, B are equipollent if there is a bijection f : A B. Note: If A is equipollent to B then B is equipollent to A. A set A is called denumerable if it is equipollent with N. A set A is called countable if it is either finite or denumerable.
17 N is denumerable. Z is denumerable. The set of even integers is denumerable. The set {5 + 1/n n N} is denumerable.
18 (a) Any infinite subset of N is denumerable. (b) Any infinite subset of a denumerable set is denumerable.
19 (a) N N is denumerable. Use f : N N N, f((a, b)) = 2 a 1 (2b 1). (b) Q + is denumerable. Use f : Q + N N, f(p/q) = (p, q) assuming p, q have no common factors.
20 A set A is called uncountable if it is not countable. Theorem 2 [Cantor s Theorem] The set of real numbers is uncountable. Georg Cantor, ,
21 Proof. Can you enumerate all possible infinite strings of blue/red marbles?
22 Theorem 3 For every set X, X and P(X) are not equipollent.
23 Proof. Show there is no surjection from X to P(X). By contradiction suppose f : X P(X) is a surjection. Let Y = {x X x / f(x)}. Let y X be such that f(y) = Y. If y f(y) = Y then by definition of Y, y / Y. If y / f(y) = Y then be definition of Y, y Y.
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