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Let $S=\{a,b,c\}\;$ and$ \;T = \{1,2,3\}$. Find the inverse of the following function \(F\) from \(S\) to \(T\), if it exists - \[\;\; F=\{(a,3), (b,2), (c,1)\}\]

Note: This is part 1 of a 2 part question, split as 2 separate questions here.

1 Answer

  • A function $g:T \to S$ if one-ne and onto is the inverse of $f:S \to T$ for every element.
  • For finite sets, if $(a,b) \in f \rightarrow (b,a) \in f^{-1}$
Given $S=\{a,b,c\};T=\{1,2,3\}$ and the function $F=\{(a,3),(b,2),(c,1)\}$.
$\Rightarrow F(a)=3\qquad F(b)=2\qquad F(c)=1$
Since $F(a),F(b), F(c)$ are distinct, $F$ is both one-one and onto, $\rightarrow$ $F^{-1} exists$
We define $F^{-1}:T \to S$ by $ F^{-1}=\{(3,a),(2,b),(1,c)\}$
answered Mar 20, 2013 by balaji.thirumalai

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