Let $f:\{1,2,3\}\to \{a,b,c\}$ be one-one and onto function given by $f\left(1\right)=a,f\left(2\right)=b$ and $f\left(3\right)=c$. Show that there exists a function $g:\{a,b,c\}\to \{1,2,3\}$ such that $gof=I_x$ and $fog=I_p$ where, $X=\{1,2,3\}$ and $Y=\{a,b,c\}$.

Q. Consider $f:\{1,2,3\}\to \{a,b,c\}$ and $g:\{a,b,c\}\to \left\{\text{apple, ball, cat}\right\}$ defined as $f\left(1\right)=a,f\left(2\right)=b,f\left(3\right)=c,g\left(a\right)=apple,g\left(b\right)=ball$ and $g\left(c\right)=cat$. Show that $f.g$ and $gof$ are invertible. Find out $f^{-1},g^{-1}$ and $(gof{)}^{-1}$ and show that $(gof{)}^{-1}=f^{-1}0g^{-1}$.

Q.

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f⁻¹ and show that (f⁻¹)⁻¹ = f.

Q. Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f⁻¹, g⁻¹ and gof⁻¹ and show that (gof)⁻¹ = f ⁻¹o g⁻¹.

Q. Which of the following functions from A to B are one-one and onto?
(i) f₁ = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}
(ii) f₂ = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
(iii) f₃ = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}

Q. Let $A=\{1,2,3\},B=\{4,5,6,7\}$ and let $f=\{(1,4),(2,5),(3,6)\}$ be a function from $A$ to $B$.Show that $f$ is one-one.