VIEWS: 0 PAGES: 12 POSTED ON: 7/26/2013 Public Domain
Functions p A function f from X to Y (in symbols f : X ® Y) is a relation from X to Y such that Dom(f) = X and if two pairs (x,y) and (x,y’) Î f, then y = y’ n Example: Dom(f) = X = {a, b, c, d}, Rng(f) = {1, 3, 5} f(a) = f(b) = 3, f(c) = 5, f(d) = 1. Domain and Range n Domain of f = X n Range of f = { y | y = f(x) for some x ÎX} n A function f : X ® Y assigns to each x in Dom(f) = X a unique element y in Rng(f) Í Y. n Therefore, no two pairs in f have the same first coordinate. Modulus operator p Let x be a nonnegative integer and y a positive integer p r = x mod y is the remainder when x is divided by y Examples: 1 = 13 mod 3 6 = 234 mod 19 4 = 2002 mod 111 p mod is called the modulus operator One-to-one functions p A function f : X ® Y is one-to-one Û for each y Î Y there exists at most one x Î X with f(x) = y. p Alternative definition: f : X ® Y is one-to-one Û for each pair of distinct elements x1, x2 Î X there exist two distinct elements y1, y2 Î Y such that f(x1) = y1 and f(x2) = y2. Examples: n 1. The function f(x) = 2x from the set of real numbers to itself is one-to-one n 2. The function f : R ® R defined by f(x) = x2 is not one-to-one, since for every real number x, f(x) = f(-x). Onto functions A function f : X ® Y is onto Û for each y Î Y there exists at least one x Î X with f(x) = y, i.e. Rng(f) = Y. n Example: The function f(x) = ex from the set of real numbers to itself is not onto Y = the set of all real numbers. However, if Y is restricted to Rng(f) = R +, the set of positive real numbers, then f(x) is onto. Bijective functions A function f : X® Y is bijective Û f is one-to-one and onto n Examples: p 1. A linear function f(x) = ax + b is a bijective function from the set of real numbers to itself p 2. The function f(x) = x3 is bijective from the set of real numbers to itself. Inverse function p Given a function y = f(x), the inverse f -1 is the set {(y, x) | y = f(x)}. p The inverse f -1 of f is not necessarily a function. n Example: if f(x) = x2, then f -1 (4) = Ö4 = ± 2, not a unique value and therefore f is not a function. p However, if f is a bijective function, it can be shown that f -1 is a function. Exponential and logarithmic functions p Let f(x) = 2x and g(x) = log 2 x = lg x n f ◦ g(x) = f(g(x)) = f(lg x) = 2 lg x = x n g ◦ f(x) = g(f(x)) = g(2x) = lg 2x = x p Therefore, the exponential and logarithmic functions are inverses of each other. Composition of functions p Given two functions g : X ® Y and f : Y ® Z, the composition f ◦ g is defined as follows: f ◦ g (x) = f(g(x)) for every x Î X. q Example: g(x) = x2 -1, f(x) = 3x + 5. Then f ◦ g(x) = f(g(x)) = f(3x + 5) = (3x + 5)2 - 1 q Composition of functions is associative: f ◦ (g ◦h) = (f ◦ g) ◦ h, q But, in general, it is not commutative: f ◦ g ¹ g ◦ f. Binary operators A binary operator on a set X is a function f that associates a single element of X to every pair of elements in X, i.e. f : X x X ® X and f(x1, x2) Î X for every pair of elements x1, x2. n Examples of binary operators are addition, subtraction and multiplication of real numbers, taking unions or intersections of sets, concatenation of two strings over a set X, etc. Unary operators p A unary operator on a set X associates to each single element of X one element of X. n Examples: 1. Let X = U be a universal set and P(U) the power set of U. Define f : P(U) ® P(U) the function defined by f (A) = A', the set complement of A in U, for every A Í U. Then f defines a unary operator on P(U). String inverse Let X be any set, X* the set of all strings over X. If a = x1x2…xn Î X*, let f(a) = a-1 = xnxn- 1…x2x1, the string written in reverse order. Then f :X* ® X* is a function that defines a unary operator on X*. Observe that aa -1 = a -1a = l