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Prove the following assertions for sets A and B from an universe U without using Venn Diagrams or membership tables:
a) (10 pts) A ⊆ B if and only if A ∩ ¬B = ∅.
b) (10 pts) A ⊆ B if and only if ¬A ∪ B = U.
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It wants you to prove them without using Venn Diagrams, but that by no means prohibits you from using the diagrams to help see what's going on. It is always advisable to draw a Venn Diagram for these types of questions (as long as you know exactly what the Diagram represents).
I will give you an example of how to prove the forward direction in part a), which will hopefully give you an idea of how to approach the other 3 proofs.
a) If A ⊆ B, then for all x in A, x is in B. So if x is not in B, then x is not in A (this is just the contrapositive of the previous statement, and can 'be seen' from the appropriate Venn diagram). i.e. if x is in ¬B, then x is in ¬A. Therefore A ∩ ¬B = ∅ (as A ∩ ¬A = ∅).
Last edited by DLF24 (2011-03-06 00:41:39)
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Ty, got this one shortly after i posted. Thanx for the help though. Much appreciated!
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