Inverse Of A Function

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Formal Definition

Let $\phi : A \rightarrow B$ be a function mapping $A$ into $B$. A map $\phi^{-1} : B\rightarrow A$ is called an inverse of $\phi$ if $\phi^{-1} (\phi (x)) = x$ for all $x \in A$ and $\phi (\phi^{-1} (y)) = y$ for all $y \in B$.

Informal Definition

It is switching the ordered pair of $\phi$, $(x,y)$, to give a one-to-one function mapping $A$ onto $B$, $(y,x)$.

Example(s)

$A=\{1,2,3\}$ and $B=\{a,b\}$\Let the relation $\phi : A \rightarrow B$ be given by the ordered pairs $\{\{1,a\},\{1,b\},\{2,a\},\{2,b\},\{3,a\},\{3,b\}\}$\Then $\phi^{-1} : B \rightarrow A$ is given by the ordered pairs $\{\{a,1\},\{b,1\},\{a,2\},\{b,2\},\{a,3\},\{b,3\}\}$\
Non-example(s)

$A=\{1,2,3\}$ and $B=\{a,b\}$\Let $\phi : A \rightarrow B$ be $\{\{1,a\},\{1,b\},\{2,a\},\{2,b\},\{3,a\},\{3,b\}\}$\Then $\phi^{-1} : A \rightarrow B]] is not [[$\{\{1,a\},\{1,b\},\{2,a\},\{2,b\},\{3,a\},\{3,b\}\}\$

Additional Comments
Note that the inverse of a function, although a mapping, is not necessarily a function. For example, the inverses of functions that are not one-to-one will not be functions.

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