Thread: Prove propositions are logically equivalent

I've got these questions and I've given it some thought but I'm not how to move forward. The question is.

Let A and B be two sets. Prove the following propositions are logically equivalent.
a) B is a subset of A
b) A intersection of B = B
c) A union of B = A

I was thinking to convert this notation into proposition notation (and, or, not, etc..) and then use thruth tables to prove they are the same. The reason I cant move forward with this question this way is that I have no idea how set theory notation can be changed to proposition notation, or even it this is the right way to answer a question life this. Any help or resource you can point me to would be really great. thanks in advance.

I will do one of these for you: .
Given that . Suppose that .
From the given we have so .
Now you do .

Thanks for your willingness to help...

I'm starting to get it.. we are proving that question (a) is equivalent to question (b) and (b) is equivalent to question (c) , thus (c) is equivalent to question (a)

I was thinking that questions (a), (b) and (c) where all serparate questions and unrelated and I needed to prove each statement serparately. I have attached these 2 approaches. Which method do you think is the correct way to proceed?

I have looked at both of the attachments.
Frankly, I do not follow either.
But the problem is clearly to show that those three are equivalent.
.

well looks like I have a lot more work to do...
thank you.. :-)