Some basic definition questions of set theory

[Ed Quanta]

I have to prove the following theorem,

1) If f:A->B is a surjection, and g:B->C is a surjection then g dot f:a->C is a surjection

Well this makes sense and I am not sure how to PROVE it

Is it sufficient to say the following

if for every element b of B, there exists a element a of A such that f(a)=b and same for B->C for g(b)=c is true, Then

for every element of c in C, there must exist an a of A such that g(f(a))=c since for every b in B there exists f(a)=b, and we know g(b)= c is true for every element of c,

 

[matt grime]

Yes that's the right idea though the phrase 'we know g(b)=c is true for every..' is, erm, not what you want to write, but I think that is just the sentence structure nothing more.

You must show that for all c in C, there is an a in A such that gf(a)=c, which is what you've done.

 

[tacman]

hence it is a surjection

Yes, your explanation is correct. To prove that g∘f is a surjection, we need to show that for every element c in C, there exists an element a in A such that g(f(a))=c. And since f is a surjection, for every element b in B, there exists an element a in A such that f(a)=b. Similarly, g is a surjection, so for every element b in B, there exists an element c in C such that g(b)=c. Therefore, for every element c in C, there exists an element a in A such that g(f(a))=c, making g∘f a surjection.