Question 1001910: If |P| = 10, |Q| = 12, and |P ∪Q| = 15, what is |P ∩Q|?pls help with a little explanation.thanks in advance and more power.
Answer by ikleyn(52778)   (Show Source):
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If |P| = 10, |Q| = 12, and |P ∪Q| = 15, what is |P ∩Q|?pls help with a little explanation.thanks in advance and more power.
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If P and Q are non-intersecting sets of elements, then it would be 10 + 12 = 22
elements in their union. But there are only 15 elements in the union.
It means that P and Q have a non-empty intersection.
Let x be the number of elements in the intersection |P ∩Q|.
Let p be the number of elements of P that are NOT in Q, and
let q be the number of elements of Q that are NOT in P.
Then
p + x = 10, (1)
q + x = 12, (2)
and
p + q + x = 15 (3)
(for the union |P ∪Q|).
Add (1) and (2). You will get
p + q + 2x = 22. (4).
Now distract (3) from (4). You will get
x = 22 - 15 = 7.
Answer. There are 7 elements in the intersection of P and Q: |P ∩Q| = 7.
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