Question

If $A=\{1,2,3\},B=\{\alpha ,\beta ,\lambda \}C=\{p,q,r\}$ and $f:A\to B,g:B\to C$ are defined by $f=\left\{\left(1,\alpha \right),\left(2,\lambda \right)\left(3,\beta \right)\right\}$ and $g=\left\{\left(\alpha ,q\right),\left(\beta ,r\right),\left(\lambda ,p\right)\right\}$ Show that $f$ and $g$ are bijective functions and ${\left(gof\right)}^{-1}=f^{-1}o{g}^{-1}$

Answer 1

$A=\{1,2,3\},B=\{\alpha .\beta \},C=\{p,q,r\}$

Refer to Image 01

$f=A⟶B$

$f=\left\{\right(1,\alpha ),(2,\lambda ),(3,\beta \left)\right\}$

$g=B⟶C$

$f:A⟶B$ is clearly a bijective function.

As every element in A has unique image in B and every element of B $\exists$ unique preimage in A.

Similarly $g:B⟶C$ is also a bijective function.

$g=\left\{\right(\alpha ,q),(\beta ,r),(\lambda ,p\left)\right\}$

$gof\left(x\right)=\left\{\right(1,q),(2,p),(3,r\left)\right\}$

$(gof{)}^{-1}=\{(q,1),(p,2),(r,3)\}$

Refer to image 02

$g^{-1}=\left\{\right(q,\alpha ),(p,\lambda ),(r,\beta \left)\right\}$

$f^{-1}=\left\{\right(\alpha ,1)$$,(\lambda ,2),(\beta ,3)\}$.

$f^{-1}o{g}^{-1}=\left\{\right(q,1),(p,2),(r,3\left)\right\}$

$\because (gof{)}^{-1}=f^{-1}o{g}^{-1}$