# Cartesian Proof Sets

• Mar 4th 2009, 09:00 PM
jacob233
Cartesian Proof Sets
Prove:

A × B = ∅ if and only if A = ∅ or B = ∅.
• Mar 4th 2009, 10:15 PM
kalagota
$\displaystyle |A\times B| = |A|\cdot |B|$
• Mar 4th 2009, 10:31 PM
Jhevon
Quote:

Originally Posted by jacob233
Prove:

A × B = ∅ if and only if A = ∅ or B = ∅.

i would do this by proving the forward direction using the contrapositive, and the converse via a direct proof.

at some point in your text you should have it written that $\displaystyle A \times \emptyset = \emptyset$ for every set $\displaystyle A$. this should help you prove the converse directly.

for the forward direction, assume NOT $\displaystyle A = \emptyset$ or $\displaystyle B = \emptyset$. then by DeMorgan's laws, we have that $\displaystyle A \ne \emptyset$ AND $\displaystyle B \ne \emptyset$. now if both $\displaystyle A$ and $\displaystyle B$ are nonempty, what can you say about $\displaystyle A \times B$?
• Mar 5th 2009, 04:51 PM
kalagota
here is another hint: it should be trivial that $\displaystyle |A| = 0 \Longleftrightarrow A=\emptyset$

use this hint and the previous hint i posted and you shall have the (two way) proof i am thinking..