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#1 2007-10-29 01:53:53

xXxEmZyxXx
Member
Registered: 2007-10-15
Posts: 12

functions

let a,b be sets
suppose there is a function g:B->A with the property that g(f(a))=a for all aEA. show that  f has to be surjective

please help!!!

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#2 2007-10-29 05:28:47

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: functions

Hmm, let me phrase the question more clearly.

“Let A, B be sets and f:AB be a function. Suppose there is a function g:BA with the property that g(f(a)) = a for all aA. Show that if g is injective f has to be surjective.”

Let bB. Let c = g(b). Then cA and by the property of g, g(f(c)) = c = g(b). Since g is injective, f(c) = b; hence f is surjective.

NB: g must be injective, otherwise it won’t work. For example, if A = {x}, B = {1,2}, f(x) = 1, g(1) = g(2) = x, then f is not surjective because there is no aA such that f(a) = 2.

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