Cartesian Product

brudman · #1 $Old$ November 13th, 2009, 08:15 AM

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?

Plato · #2 $Old$ November 13th, 2009, 08:25 AM

Quote:

Originally Posted by brudman $View Post$

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 $A \times B = \emptyset ~?$

Give it a try!

brudman · #3 $Old$ November 13th, 2009, 08:59 AM

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

Plato · #4 $Old$ November 13th, 2009, 09:11 AM

Quote:

Originally Posted by brudman $View Post$

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

Well at least one of the two would have to be $\emptyset$ .

brudman · #5 $Old$ November 13th, 2009, 10:14 AM

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

Plato · #6 $Old$ November 13th, 2009, 10:20 AM

Quote:

Originally Posted by brudman $View Post$

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.

brudman · #7 $Old$ November 15th, 2009, 09:23 AM

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?