Discrete Mathematics
# Principle of Inclusion and Exclusion

If $A$ and $B$ are two sets such that $\lvert{A \cup B}\rvert=45, \lvert{A \cap B}\rvert=21 \text{ and } \lvert{A \setminus B}\rvert=11,$ then what is the value of $\lvert{B}\rvert?$

**Details and assumptions**

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Calvin has 23 books on his shelf, all of which are either Mathematics books or hardcover books. If 18 of the 23 are hardcover books, and 12 of the 23 are Mathematics books, how many hardcover Mathematics books does Calvin have on his shelf?

**Details and assumptions**

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If set $A$ contains $57$ elements, set $B$ contains $316$ elements and $A \cap B$ contains $15$ elements, how many elements are contained in $A \cup B?$

**Details and assumptions**

You may choose to read the summary page Set Notation.