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# State with reason whether following functions have inverse: (iii) $h: \{2,3,4,5,\} \to \{7,9,11,13\}$ with $h=\{(2,7),(3,9),(4,11),(5,13) \}$

This is part of a multi-part question on Clay6 and is answered here separately.

Toolbox:
• To check if a function is invertible or not ,we see if the function is both one-one and onto.
• A function $f: X \rightarrow Y$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function.
• A function$f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
Given function h defined by $h:\{2,3,4,5\} \to \{7,9,11,12,13\}$
Step1: Checking for Injective or One-One function:
From the given definition of $h=\{(2,7),(3,9),(4,11)(5,13)\}$, we can see that it is one-one, as for every ${2,3,4,5} \in h,\;$ all have distinct images under $h$, i.e., $h(2) \neq h(3) \neq h(4) \neq h(5)$.
Therfore, the function is one-one. We need to now check further for onto.
Step2: Checking for Surjective or On-to:
From the given definition of $h=\{(2,7),(3,9),(4,11)(5,13)\}$, we can see that every element in $\{7,9,11,13\}$ then exists an element $\{2,3,4,5\}$ such that $h(x)=y$ is onto.
$\rightarrow$ The pre image of 7 is 2 , the pre image of 9 is 3, the pre image of 11 is 4 and pre image of 13 is 5
Therfore, the function is onto as well as one-one and hence is invertible.
edited Mar 19, 2013