Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ given by $f(1)=a, f(2)=b$ and $f(3)=c$. Find $f^{-1}$ and show that $\left(f^{-1}\right)^{-1}=f$.

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written 2.9 years ago by

prajapatijaimin • 3.3k

• modified 2.9 years ago

Solution:

If we define $g:\{a, b, c\} \rightarrow\{1,2,3\}$ as $g(a)=1, g(b)=2, g(c)=3$ , then we have:

$\begin{aligned} &(f \circ g)(a)=f(g(a))=f(1)=a \\\\ &(f \circ g)(b)=f(g(b))=f(2)=b \\\\ &(f \circ g)(c)=f(g(c))=f(3)=c \\ \end{aligned}$

And,

$\begin{aligned} &(g \circ f)(1)=g(f(1))=g(a)=1 \\\\ &(g \circ f)(2)=g(f(2))=g(b)=2 \\\\ &(g \circ f)(3)=g(f(3))=g(c)=3 \\ \end{aligned}$

$ \therefore …

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