written 2.9 years ago by | • modified 2.9 years ago |
Solution:
If we define g:{a,b,c}→{1,2,3} as g(a)=1,g(b)=2,g(c)=3, then we have:
(f∘g)(a)=f(g(a))=f(1)=a(f∘g)(b)=f(g(b))=f(2)=b(f∘g)(c)=f(g(c))=f(3)=c
And,
(g∘f)(1)=g(f(1))=g(a)=1(g∘f)(2)=g(f(2))=g(b)=2(g∘f)(3)=g(f(3))=g(c)=3
∴g∘f=IX
and
f∘g=IY
where
X={1,2,3}
and
Y={a,b,c}.
Thus, the inverse of f exists and f−1=g.
∴f−1:{a,b,c}→{1,2,3} is given by,
f−1(a)=1,f−1(b)=2,f1(c)=3
Let us now find the inverse of f−1 i.e., find the inverse of g.
If we define h:{1,2,3}→{a,b,c} as
h(1)=a,h(2)=b,h(3)=c, then we have:
(g∘h)(1)=g(h(1))=g(a)=1(g∘h)(2)=g(h(2))=g(b)=2(g∘h)(3)=g(h(3))=g(c)=3
And,
(h∘g)(a)=h(g(a))=h(1)=a(h∘g)(b)=h(g(b))=h(2)=b(h∘g)(c)=h(g(c))=h(3)=c
∴goh=IX and hog=Iγ, where X={1,2,3} and Y={a,b,c}.
Thus, the inverse of g exists and g−1=h⇒(f−1)−1=h.
It can be noted that h=f.
Hence,
(f−1)−1=f.