If A= {1, 2, 3}, B = {alpha,beta, m}, C = {p, q, r}, f: A → B and g: B→C are defined by, f = {(1, alpha), (2, m), (3, beta)} g= {(alpha, q), (beta, r), (m, p)} then show that (gof)-1 = f-1og-1.
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Answer:A={1,2,3},B={α.β},C={p,q,r}
Refer to Image 01
f=A⟶B
f={(1,α),(2,λ),(3,β)}
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 ∃ unique preimage in A.
Similarly g:B⟶C is also a bijective function.
g={(α,q),(β,r),(λ,p)}
gof(x)={(1,q),(2,p),(3,r)}
(gof)
−1
={(q,1),(p,2),(r,3)}
Refer to image 02
g
−1
={(q,α),(p,λ),(r,β)}
f
−1
={(α,1),(λ,2),(β,3)}.
f
−1
og
−1
={(q,1),(p,2),(r,3)}
∵(gof)
−1
=f
−1
og
−1
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