Axiom of Pair and Axiom of Union?

[A.MHF]

So I've been learning Set Theory by myself through Jech and Hrabeck textbook, and I'm having trouble understanding some axioms.

1. Homework Statement
What exactly is the difference between the axiom of pair and axiom of union?
From what I understood, the axiom of pair tells us that there is a set C whose elements are the elements of A and the elements of B. As for the axiom of union, what I understood is that it tells us that each member in a set is a set itself. (Please correct me if I'm wrong). However, I've seen proofs using axiom of union to prove that there is a set that exists which has the exact elements of both set A and set B. But if that's what the axiom is for, then what's the use of axiom of pair? Please help me understand the role of each of those axioms.

Homework Equations

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The Attempt at a Solution

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[micromass]

Do you know the difference between $\{A,B\}$ and $A\cup B$?

 

[A.MHF]

Do you know the difference between $\{A,B\}$ and $A\cup B$?

The first one is a set whose elements are A and B. The second one is a set whose elements are the elements of the sets A and B.
Is that correct?

 

[micromass]

Right. The axiom of pair just says $\{A,B\}$ exists. The axiom of union says $A\cup B$ exists (well, more or less). So they're very different statements.

 

[A.MHF]

Right. The axiom of pair just says $\{A,B\}$ exists. The axiom of union says $A\cup B$ exists (well, more or less). So they're very different statements.

I see. Just to be clear, is this right:
Let's say there is are sets A:{1,2,3} and B:{4,5,6}.
The axiom of pair would tell me that this set exists: {{123},{456}}. The axiom of union would tell me that this one exists: {1,2,3,4,5,6}.

 

[micromass]

Yes.