Question:
If $A$ and $B$ are two sets such that $n(A)=37, n(B)=26$ and $n(A \cup B)=51$, find $\mathbf{n}(\mathbf{A} \cap \mathbf{B})$
Solution:
Given:
$n(A)=37$
$n(B)=26$
$n(A \cup B)=51$
To Find: $n(A \cap B)$
We know that,
$|A \cup B|=|A|+|B|-|A \cap B|$ (where $A$ and $B$ are two finite sets)
Therefore,
$n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$51=37+26-n(A \cap B)$
$n(A \cap B)=63-51=12$
Therefore,
$n(A \cap B)=12$
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