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

Question

If $A = \{1, 2, 3\}$, $B = \{\alpha, \beta, \gamma\}$, $C = \{p, q, r\}$ and $f : A 	o B$, $g : B 	o C$ are defined by $f = \{(1, \alpha), (2, \gamma), (3, \beta)\}$, $g = \{(\alpha, q), (\beta, r), (\gamma, p)\}$, then show that f and g are bijective functions and $(gof)^{-1} = f^{-1}og^{-1}$

Accepted Answer

>

> [!TLDR]
> Check each of $f$ and $g$ is one-one and onto (hence bijective), compute $gof$ and its inverse directly, then compute $f^{-1}og^{-1}$ and show the two results are identical.
>
> **Given:** $A=\{1,2,3\}$, $B=\{\alpha,\beta,\gamma\}$, $C=\{p,q,r\}$, $f=\{(1,\alpha),(2,\gamma),(3,\beta)\}$, $g=\{(\alpha,q),(\beta,r),(\gamma,p)\}$.
>
> **Step 1. $f$ is bijective.** The images $\alpha,\gamma,\beta$ of $1,2,3$ are all distinct $\Rightarrow$ $f$ is one-one (injective). Every element of $B=\{\alpha,\beta,\gamma\}$ appears as an image $\Rightarrow$ $f$ is onto (surjective). Hence $f$ is bijective.
>
> **Step 2. $g$ is bijective.** The images $q,r,p$ of $\alpha,\beta,\gamma$ are all distinct $\Rightarrow$ $g$ is one-one. Every element of $C=\{p,q,r\}$ appears as an image $\Rightarrow$ $g$ is onto. Hence $g$ is bijective.
>
> **Step 3. Compute $gof:A	o C$:**
> $$gof(1)=g(f(1))=g(\alpha)=q,\quad gof(2)=g(f(2))=g(\gamma)=p,\quad gof(3)=g(f(3))=g(\beta)=r$$
> $$gof = \{(1,q),(2,p),(3,r)\}$$
>
> **Step 4. Compute $(gof)^{-1}$** (swap each pair):
> $$(gof)^{-1} = \{(q,1),(p,2),(r,3)\}$$
>
> **Step 5. Compute $f^{-1}$ and $g^{-1}$:**
> $$f^{-1} = \{(\alpha,1),(\gamma,2),(\beta,3)\}, \qquad g^{-1} = \{(q,\alpha),(r,\beta),(p,\gamma)\}$$
> …

🎓Unlock everything free for 14 days

Unlock everything free for 14 days