Question

let f:A-->B, g:B-->C be bijection,then S.T gof: A--->C is a bijection​

Answer 1

Answer:

f : A → B, g : B → C are bijections ⇒ gof : A → C is a bijection Also g–1 : C → B and f–1 : B → A are bijections ⇒ f –1og–1 : C → A is a bijection. Let c be any element of C. Then ∃ an element b ∈ B such that g(b) = c ⇒ b = g–1(c) Also ∃ an element a ∈ A such that f(a) = b ⇒ a = f–1 (b) Now (gof) (a) = g(f(a)) = g(b) = c ⇒ a = (gof)–1 (c) ⇒ (gof)–1 (c) = a ——— (1) Also (f–1og–1) (c) = f–1(g–1(c)) =f–1(b) = a ——— (2) ∴ From (1) and (2); (gof)–1(c) = (f–1og–1) ( c) ⇒ (gof)–1 = f–1og–1.Read more on Sarthaks.com - https://www.sarthaks.com/533135/let-f-a-b-g-b-c-be-bijections-then-prove-that-gof-1-f-1og-1?show=533143#a533143