Cartesian Proof Sets

**jacob233** · March 4th 2009, 09:00 PM

Prove:

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

**kalagota** · March 4th 2009, 10:15 PM

maybe this will help you..
$|A\times B| = |A|\cdot |B|$

**Jhevon** · March 4th 2009, 10:31 PM

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 $A \times \emptyset = \emptyset$ for every set $A$ . this should help you prove the converse directly.

for the forward direction, assume NOT $A = \emptyset$ or $B = \emptyset$ . then by DeMorgan's laws, we have that $A \ne \emptyset$ AND $B \ne \emptyset$ . now if both $A$ and $B$ are nonempty, what can you say about $A \times B$ ?

**kalagota** · March 5th 2009, 04:51 PM

here is another hint: it should be trivial that $|A| = 0 \Longleftrightarrow A=\emptyset$

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

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