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Subset Proof Questions?
Hi everyone, would really appreciate some help with these questions, they seem intuitive but I really have no idea how to work them.
1. Given that A is a subset of B, prove that AnC is a subset of BnC
2. Given that AnB' is a subset of B, prove A is a subset of B
3. Given that A is a subset of B, prove that AnB' is a subset of B
Would really appreciate help on this because this is in for tomorrow and none of my books are any help!
Thank you, all answers will get a thumbs up!
1 Answer
- Randy PLv 710 years agoFavorite Answer
Proving X is a subset of Y means proving that "If x is an element of X, then x is an element of Y".
1. Let x be an element of AnC. That means it's an element of A and it's an element of C. You want to show it's an element of B and an element of C. How would you argue it's an element of B?
These last two only make sense to me if AnB' is the empty set. That's the only way AnB' can be a subset of B.
2. Let x be an element of A... Hmmm, this one is a little trickier.
OK, how about this. Suppose x is not an element of B. Then it's an element of AnB'. But since AnB' is a subset of B, then x is an element of B. Contradiction.
3. Let x be an element of AnB'. Then it's an element of A, and it's not an element of B. But A is a subset of B, so x is an element of B. Contradiction. The contradiction came from the original assumption "Let x be an element of AnB'". So there is no such x.