# Thread: A u b = ∅ <--> a=b=∅

1. ## A u b = ∅ <--> a=b=∅

I think I'm on the right track, it seems like such an easy proof but oh well!

Anyway here's what I got so far.

Suppose A=B=∅

= ∅

= ∅ U ∅

= A U B

(I think that's a proof, atleast I think that's how my professor did it)

Suppose A U B = ∅

Case 1: x is in A

x is in A, x is not in B

Therefore A U B <--> ∅

[(for all x in (A U B), x is in ∅) And (for all x in ∅, x is in (A U B)]

Since x is in a, in order for A U B = ∅

A = ∅

Case 2: x is in b

etc.

Im really confused, that doesnt seem to prove anything, I think I'm doing something wrong, any help?

2. Originally Posted by glover_m
I think I'm on the right track, it seems like such an easy proof but oh well!

Anyway here's what I got so far.

Suppose A=B=∅

= ∅

= ∅ U ∅

= A U B

(I think that's a proof, atleast I think that's how my professor did it)
ok, this direction seems ok

Suppose A U B = ∅

Case 1: x is in A

x is in A, x is not in B

Therefore A U B <--> ∅

[(for all x in (A U B), x is in ∅) And (for all x in ∅, x is in (A U B)]

Since x is in a, in order for A U B = ∅

A = ∅

Case 2: x is in b

etc.
i don't like this proof, it doesn't seem valid. it all starts from the red line. how does that follow from anything you've said. the line directly below that makes no sense either. it's just a faulty proof.

for the second direction, use the contrapositive. assume either A or B is not empty, then show that this means AUB is not empty either, pretty easy

3. Originally Posted by glover_m
I think I'm on the right track, it seems like such an easy proof but oh well!

Anyway here's what I got so far.

Suppose A=B=∅
I can only see a square on my internet reader. Is that supposed to be $\phi$, for the empty set?

= ∅

= ∅ U ∅

= A U B

(I think that's a proof, atleast I think that's how my professor did it)[/quote]
Yes, that's good: if A and B are both empty then their union is empty.
You might also do it as a "proof by contradiction": Suppose x is in A U B. Then x is in A or x is in B. But both A and B are empty either way we have a contradiction.

Suppose A U B = ∅

Case 1: x is in A

x is in A, x is not in B
No. That's not necessarily true.

Therefore A U B <--> ∅
and this certainly doesn't follow! It doesn't even make sense.
You can, of course, immediately say "If x is in A then x is in A U B". What follows from that?

[(for all x in (A U B), x is in ∅) And (for all x in ∅, x is in (A U B)]

Since x is in a, in order for A U B = ∅

A = ∅

Case 2: x is in b

etc.

Im really confused, that doesnt seem to prove anything, I think I'm doing something wrong, any help?