# Prove propositions are logically equivalent

• Mar 1st 2011, 03:27 AM
spaz84
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.
• Mar 1st 2011, 04:32 AM
Plato
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 $a \Rightarrow b~\&~b \Rightarrow c$.
• Mar 1st 2011, 02:43 PM
spaz84
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?

Attachment 21006
Attachment 21005
• Mar 1st 2011, 03:29 PM
Plato
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.
$a\to b\to c\to a$.
• Mar 1st 2011, 03:33 PM
spaz84
well looks like I have a lot more work to do...
thank you.. :-)