
OneOne Function A function f: A>B is called a oneone function if distinct elements of A have distinct images in B, i.e if a_{1}, a_{2} €A and a_{1} ≠a_{2}=>f(a_{1}) ≠ f(a_{2}) Equivalently, we say f:A > B is oneone if and only if for all a_{1}, a_{2} € A, f(a_{1}) = f(a_{2}) => a_{1} = a_{2}. Note 1: A oneone function is also called an injective function or injection. Note 2: If A and B are finite sets and f: A > B is injective. Then n(A)<= n(B). Note 3: If n(A) = P and n(B) = q, then the number of possible mappings from A to B is q_{p}. Illustrations: If A = {4, 5, 6} and B = {a, b, c, d} and if A > B such that f = {(4,a), (5,b), (6,c)}, then f is oneone. The mapping f: R>R such that f(x) = x^{2} is not a oneone function since f(2) = 4 and f(2) = 4, that is two distinct elements 2 and 2 have the same image 4.
Example: Find if the following functions are oneone or not.
Solution: 1. Let x, y be two arbitrary elements of domain f, (x,y € R) such that f(x) = f(y). Then f(x) = f(y) =>x^{3} = y^{3} =>x = y 2. Let x, y be two arbitrary elements of Z such that f(x) = f(y). Then f(x) = f(y) =>x^{2} +5 = y^{5} + 5 =>x^{2} = y^{2} =>x = + or  y. Since f(x) = f(y) does ot yield a unique answer and x = y but gives x = + or  y, so f is not a oneone function. Suppose we have f(2) and f(2), for either cases we get 9, thus two distinct elements 2 and 2 have the same image. Hence f is oneone function.
Example: Directions: Solve the following problems. Also write at least 5 examples of your own. 