G 1 in the codomain does not have a preimage in the

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Unformatted text preview: A function f is called a surjection if it is onto. NO! YES! YES! Bijec$ons Definition: a function f is a bijection (one- to- one correspondence), if it is both one- to- one and onto, i.e., both surjective and injective. A B a x b y c d YES! z w NO! surjection but not injection Showing that f is one- to- one or onto Example 1: let f be a function from { a, b, c, d } to { 1, 2, 3 } defined by f (a) = 3, f (b) = 2, f (c) = 1, and f (d) = 3. Is f an onto function? Solution: Yes, f is onto since all three elements of the codomain are images of elements in the domain. If the codomain were changed to { 1, 2, 3, 4 }, f would not be onto. Example 2: is the following function f (x): Z → Z, where Z is the set of integers and f (x) = x2 onto? Solution: No, f is not onto because, e.g., −1 in the codomain does not have a preimage in the domian. Inverse Func$ons Definition: let f be a bijection from A to B. Then the inverse of f, denoted by f −1, is the function from B to A defined as —༉  No inverse exists unless f is a bijection. Inverse Func$ons A a f B V b A a b W c d B V W c X Y X d Y Ques$ons Example 1: let f be the function from { a, b, c } to {1, 2, 3 } such that f (a) = 2, f (b) = 3, and f (c) = 1. Is f invertible and if so, what is its inverse? Solution: the function f is invertible because it is a o...
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This document was uploaded on 02/27/2014 for the course CS 215 at SIU Carbondale.

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