1. ## Cartesian Product

Suppose that A and B are sets, and that A X B is the empty set.
How can you prove that given any set C, either A is a subset of C or B is a subset of C?

2. Originally Posted by brudman
Suppose that A and B are sets, and that A X B is the empty set.
How can you prove that given any set C, either A is a subset of C or B is a subset of C?
Surely you can can show effort on this one.
Under what conditions is it ever true that $\displaystyle A \times B = \emptyset ~?$

Give it a try!

3. Wouldn't A and B have to be empty sets themselves?

4. Originally Posted by brudman
Wouldn't A and B have to be empty sets themselves?
Well at least one of the two would have to be $\displaystyle \emptyset$.

5. So how can i write a formal proof without using algebra? or do i have to use algebra?

6. Originally Posted by brudman
So how can i write a formal proof without using algebra? or do i have to use algebra?
Just write it up in sentence/paragraph form.
The empty set is a subset of every set.
So you are done.

7. So the proof will go something like this:

Suppose that A and B are sets, and that A X B is the empty set.
Let A = ∅ then ∅ × B = ∅ by definition, so A is a subset of C.
Let B = ∅ then A × ∅ = ∅ by definition, so B is a subset of C.

is this good enough?