Question

Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :

f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.

Show that f and g both are bijections and find fog and gof.

Answer 1

Proving f is a bijection:
f = {(a, v), (b, u), (c, w)} and f : A → B
Injectivity of f: No two elements of A have the same image in B.
So, f is one-one.
Surjectivity of f: Co-domain of f = {u v, w}
Range of f = {u v, w}
Both are same.
So, f is onto.
Hence, f is a bijection.

Proving g is a bijection:
g = {(u, b), (v, a), (w, c)} and g : B → A
Injectivity of g: No two elements of B have the same image in A.
So, g is one-one.
Surjectivity of g: Co-domain of g = {a, b, c}
Range of g = {a, b, c}
Both are the same.
So, g is onto.
Hence, g is a bijection.

Finding fog:
Co-domain of g is same as the domain of f.
So, fog exists and fog : {u v, w} → {u v, w}
$$\begin{gathered} \left(fog\right)\left(u\right)=f\left(g\left(u\right)\right)=f\left(b\right)=u \\ \left(fog\right)\left(v\right)=f\left(g\left(v\right)\right)=f\left(a\right)=v \\ \left(fog\right)\left(w\right)=f\left(g\left(w\right)\right)=f\left(c\right)=w \\ \mathrm{So},fog=\left\{\left(u,u\right),\left(v,v\right),\left(w,w\right)\right\} \end{gathered}$$

Finding gof:
Co-domain of f is same as the domain of g.
So, fog exists and gof : {a, b, c} → {a, b, c}
$$\begin{gathered} \left(gof\right)\left(a\right)=g\left(f\left(a\right)\right)=g\left(v\right)=a \\ \left(gof\right)\left(b\right)=g\left(f\left(b\right)\right)=g\left(u\right)=b \\ \left(gof\right)\left(c\right)=g\left(f\left(c\right)\right)=g\left(w\right)=c \\ \mathrm{So},gof=\left\{\left(a,a\right),\left(b,b\right),\left(c,c\right)\right\} \end{gathered}$$

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