So there is a similiar question in the archives which I looked at after I attempted my proof: Proving that for any sets $A,B,C$, and $D$, if $(A\times B)\cap (C\times D)=\emptyset $, then $A \cap C = \emptyset $ or $B \cap D = \emptyset $

But it is not exactly the same, so I wanted to write my proof out from start to finish to see if my thought process was correct.

1) First I just experimented with some sets to see if anything came about: let $A = \{1\}$, $B = \{2\}$, $C = \{3\}$, $D = \{4\}$:

$(A\times B) \cap (C\times D)$ side:

$$(1,2)\cap (3,4) = (\emptyset,\emptyset)$$ [not sure if i could state this, but it is what I said in my solution]

$(A\cap C)\ \times (B\cap D)$ side: $$\emptyset\ \times \emptyset = (\emptyset, \emptyset)$$

Ok so I established what appears to be equality, so now I have to prove it.

let $(x,y) \in (A\ X\ B) \cap (C\ X\ D)$ --> $(x\in A \cap y\in B) \cap (x\in C \cap y\in D)$ --> $(x\in A \cap x\in C) \cap (y\in B \cap y\in D)$ --> $x\in (A\cap C)\ X\ y\in (B\cap D)$ --> $ (x,y)\in (A\cap C)\ X\ (B\cap D)$ Done.

Then I would have to do the other way as well but it would amount to a similar argument.

P.S: How to get lines of my proof to line up with arrows?

  • 1
    $\begingroup$ Your experiment is not really senseful -- $A, B, C, D$ are supposed to be sets, not numbers. $\endgroup$ – hmakholm left over Monica May 5 '15 at 19:42
  • $\begingroup$ Please use \times for $\times$ $\endgroup$ – Gregory Grant May 5 '15 at 19:43
  • $\begingroup$ I treated them as singleton sets. I just don't know how to write the squigly bracket in latex $\endgroup$ – dc3rd May 5 '15 at 19:43
  • $\begingroup$ @dc3rd type " \{ " and " \} " $\endgroup$ – user265675 May 5 '15 at 19:46
  • 2
    $\begingroup$ Somewhat related and may help. $\endgroup$ – Daniel W. Farlow May 5 '15 at 19:50

$$(x,y)\in (A\times B)\cap (C\times D) \iff \left\{\begin{array}{c}(x,y)\in A\times B \\ (x,y) \in C\times D \end{array}\right. \iff \left\{\begin{array}{c}x \in A,\ y\in B \\ x \in C,\ y\in D \end{array}\right. \iff$$

$$\iff \left\{\begin{array}{c}x \in A,\ x\in C \\ y \in B,\ y\in D \end{array}\right. \iff \left\{\begin{array}{c}x \in A\cap C \\ y \in B\cap D \end{array}\right. \iff (x,y) \in (A\cap C)\times (B\cap D)$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.