Math Is Fun Forum

xXxEmZyxXx

Member

Registered: 2007-10-15

Posts: 12

injective function

let a b be sets 
f:A->B
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 injective

please help!!!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: injective function

If you assume that f is not injective, you should be able to reach a conclusion that g is not well defined.  To rephrase this proof in a way that makes sense, it is saying that if f has an inverse, then f is injective.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

xXxEmZyxXx

Member

Registered: 2007-10-15

Posts: 12

Re: injective function

the answer should not refer to the inverse of f since it may not exist

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: injective function

If f is injective, then it has an inverse.  What you are doing is showing that g(f(a))=a for all aEA. => f is injective => f has an inverse.  So you aren't allowed to assume f has an inverse, no.  But this is what the statement is saying.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: injective function

the answer should not refer to the inverse of f since it may not exist

This sounds like a comment your teacher made when he/she was marking your homework.

I suspect that you’ve been posting your homework here and copying our solutions for your homework. Do you understand the solutions, though? It is one thing to copy answers and getting marks for them – it is far more important to understand the solutions.