Set Theory Proof: A∩B=Ø implies C∩D=Ø

[dainty77]

Homework Statement

Hey guys!

I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem:

Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø.

This is a biconditional so I have to prove it both ways correct?

Any help would be greatly appreciated!

Homework Equations

The Attempt at a Solution

 

[D H]

Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø.

Where's the biconditional? I only see an if then, not an if and only if. What work have you done?As stated, the conjecture is false. Are you sure you have the sense correct in terms of which sets are subsets of some other set?

 

[dainty77]

My mistake, it is an "if then" statement.

 

[D H]

As I previously said, the conjecture as written is false. For example, consider sets of fruits. Let A={apple}, B={banana}, C={apple, orange}, and D={banana, orange}. With this, A⊆C, B⊆D, and A∩B=Ø, but C∩D={orange}, which is not the null set.

 

[dainty77]

Wow, the example of using fruit really helped clarify it a lot more. Thank you so much!