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When $P$ is a subset of $Q$, use logical connectives to

  1. Prove $P \cup Q = Q$.

  2. Prove $P \cap Q = P$.

I know that they are true, but I don't know how to use the definitions of union and intersection in terms of logical connectives to prove them.

Thanks in advance!

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One half of your question was asked here: Prove that $A \subset B$ if and only if $A \cap B = A$. Several other equivalent conditions were given here: Statements equivalent to $A\subset B$ –  Martin Sleziak Sep 30 '13 at 15:42

1 Answer 1

up vote 1 down vote accepted

Have you tried to break this down and look at it element by element?

If P is a subset of Q, then by definition, every element of P is an element of Q. So, as Q contains P, every element in either P or Q is in Q, so the union is Q.

For the second, every element of P and Q is in P, so the intersection is P.

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I know that they are true, but i don't know how to use definitions of union and intersection in terms of logical connectives to prove them –  rMath Sep 30 '13 at 13:16
    
I think you'd find that my answer actually contains those answers. Union works like 'or', intersection works like 'and'. If an element is in P and it is in Q, then it's in the intersection, etc. –  Vernepator Cur Sep 30 '13 at 13:32

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