SOLUTION: If n(A)=13, n(A∪B)=19, and n(B)=15, what is n(A∩B)? ​ Please explain .Thank you

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Question 1055232: If n(A)=13, n(A∪B)=19, and n(B)=15, what is n(A∩B)? ​


Please explain .Thank you
Answer by ikleyn(52793)   (Show Source): You can put this solution on YOUR website!
.
n(A∩B) = n(A) + n(B) - n(A∪B) = 13 + 15 - 19 = 9.

Proof

It is easier to prove an equivalent equality

n(A∪B) = n(A) + n(B) - n(A∩B).

Let us try to count all elements in A∪B.
As a first approximation, we will take n(A) + n(B).
But doing so, we count the elements of the intersection n(A∩B) twice.
So, we need distract n(A∩B) from n(A) + n(B), and in this way we get the exact number of elements in A∪B.

The proof is completed.


It is a classic problem of elementary set theory.

See also the lesson
    - Counting elements in sub-sets of a given finite set
in this site.


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Miscellaneous word problems".


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