Question

let f:A-B,g:B-C be bijection,show that gof:A-B is a bijection​

Answer 1

f:A→Bf:A→B and g:B→Cg:B→C are both one-to-one functions.

Suppose a1,a2∈Aa1,a2∈A such that (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2)

⇒g(f(a1))=g(f(a2))⇒g(f(a1))=g(f(a2)) (definition of composition) Since gg is one-to-one, therefore,

f(a1)=f(a2)f(a1)=f(a2)

And since ff is one-to-one, therefore,

a1=a2a1=a2

Thus, we have shown that if (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2) then a1=a2a1=a2

Hence, gofgof is one-to-one function.  

Answer 2

Answer:

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