# Thread: Prove propositions are logically equivalent

1. ## 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.

2. I will do one of these for you: $c \Rightarrow a$.
Given that $A\cup B=A$. Suppose that $x\in B$.
From the given we have $x\in A\cup B=A$ so $B\subseteq A$.
Now you do $a \Rightarrow b~\&~b \Rightarrow c$.

3. ## clarification

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?

4. 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.
$a\to b\to c\to a$.

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