Proving the graph of a composite function

[The1TL]

Homework Statement

Let f : A → B, g : B → C be functions where A,B,C are sets. ConsiderΓf ⊂A×B,the graph of f,Γg ⊂B×C,the graph of g. Now consider the sets Γ f ×C ⊂ A×B×C and A×Γg ⊂A×B×C. LetΓ=θ(Γf ×C∩A×Γg)⊂A×C where θ : A×B×C → A×C is the projection defined as θ((a,b,c))=(a,c). Show that Γ is the graph of g◦f.

Homework Equations

The Attempt at a Solution

I think that I should find a way to show that the first projection of gamma is a bijection. But i don't know how I can do that.

 

[The1TL]

please help I've been trying for so long

 

[HallsofIvy]

Then show us what you have been doing all that time!