# Set theoretic identities involving the Cartesian product

In "Naive Set Theory" by Halmos (amazon link) there is a chapter involving ordered pairs which eventually mentions the Cartesian product and many of it's properties. For the sake of completeness I will write the given definition here:

$A\times B = \{x\colon x=(a,b)$ for some $a$ in $A$ and for some $b$ in $B \}$. (Halmos, 24)

I have no qualms with this definition, only the problems at the end of the section. I do not know if my "proofs" are satisfactory because at times it seems like the identities are trivial. I am looking for a general method of proofing such "set theoretic identities" that works in most cases. Here is an example. Every capital letter is a set.

$(A\cup B)\times C = (A\times C)\cup(B\times C)$

Is the best method of proof showing that they are both subsets of each other or is there another way? Here is another example problem:

$(A-B)\times C = (A\times C)- (B\times C)$.

How should someone give a complete proof for these statements?

• I'd go by showing that they are subsets of each other. – user122283 May 18 '14 at 15:11
• These questions have been through the text-grinder that is MSE for so long that it's impossible to believe this is not a duplicate. Take a look at other threads which discuss this question, some include full proofs, others hints and guidance. For what it's worth, if you already have one method for proving equality of sets, use it until you run into another method. There's no need to jump ahead and ask people for other methods. There's a good pedagogical reason why you are being taught a particular method when you first start with set theory, and only later you meet more methods. – Asaf Karagila May 18 '14 at 15:13
• Also, why does it say "proof verification" in the title? You haven't given any proof for us to verify. – Asaf Karagila May 18 '14 at 15:14
• @AsafKaragila Maybe I am not new to set theory. I have been exposed to the Cartesian product several times. I have taken example problems from Halmos because they are simply the canonical example that is almost always used. I see no reason why I should use subsets to prove when there are other methods such as using general theorems about the Cartesian products of unions. I am looking for proofs that exemplify the properties of sets rather than just a proof in general. I intended to include a proof when writing the title but I removed it due to length concerns and accidentally left the title. – user147887 May 18 '14 at 15:33
• n the answer to this post you can find examples of the two "basic" method to be used in order to show that A=B, with A and B "complex" set expressions: (i) prove that A⊆B and B⊆A; (ii) transform the LHS expression A through known "identities" into the RHS expression B. – Mauro ALLEGRANZA May 18 '14 at 15:53