If X and Y are two sets such that X has 40 elements, X ∪Y has 60 elements and X ∩Y has 10 elements, how many elements does Y have?

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If X and Y are two sets such that X has 40 elements, X ∪Y has 60 elements and X ∩Y has 10 elements, how many elements does Y have?

Solution:

It is known that

$n\left( X \right)\text{ }=\text{ }40$

$n(X~\cup ~Y)\text{ }=\text{ }60$

$n(X~\cap ~Y)\text{ }=\text{ }10$

This can be written as

$n(X~\cup ~Y)\text{ }=~n\left( X \right)\text{ }+~n\left( Y \right)-n(X~\cap ~Y)$

Now substitute the values

$60\text{ }=\text{ }40\text{ }+~n\left( Y \right)-10$

Calculating further

$n\left( Y \right)\text{ }=\text{ }60-\left( 40\text{ }\text{-}10 \right)\text{ }=\text{ }30$

As a result, the set Y has 30 elements

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