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Relations and Functions
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The function $g: \{ 5,6,7,8\} \to \{1,2,3,4\}$ is invertible. True or False?
related to an answer for:
State with reason whether following functions have inverse: (ii) \(g: \{ 5,6,7,8\} \to \{1,2,3,4\}\) with \(g= \{(5,4),(6,3),(7,4),(8,2)\}\)
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Mar 19, 2013
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balaji.thirumalai
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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 g defined by $g:\{5,6,7\} \to \{1,2,3,4\}$
Step1: Checking for Injective or One-One function:
From the given definition of $g=\{(5,4),(6,3),(7,4)(8,2)\}$, we can see that it is many-one, as for every ${5,6,7} \in g,\; g(5) = g(6) = g(7) = 1$.
Therfore, the function does not have an inverse. We don't need to check further for onto.
Answer: False.
answered
Mar 19, 2013
by
balaji.thirumalai
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