(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If A and B are sets, prove that a subset [itex]\Gamma\subset A X B[/itex] is the graph of some function from A to B if and only if the first projection [itex]\rho: \Gamma\rightarrow A[/itex] is a bijection.

2. Relevant equations

3. The attempt at a solution

I first thought that i should define bijection by saying that a a bijection exists when there is both injection and surjection.

If [itex]\rho: \Gamma\rightarrow A[/itex] is not a bijection then it is either

1)not surjective

2)not injective

3)both 1) and 2)

So,

I thought that i should prove that [itex]\Gamma[/itex] is not the graph of some function A -> B when the first projection is not bijective by showing the non-surjective and non-injective cases separately. And then prove that it is, in fact, [itex]\Gamma[/itex] is the graph of the function when the first projection is bijective.

So, i have to show the cases for when the first projection is not injective, when it's not surjective, and then when it is bijective.

But, how do i prove that the [itex]\Gamma[/itex] is either the graph of the function or not? in any of the 3 cases?

I just don't know where to start. Is this the right approach or is there a shorter way to do this?

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# Homework Help: Graph of a function only if first projection is bijective

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