Answer to Question #185581 in Discrete Mathematics for nellie karren

Question

If 𝐴 and B are finite sets which are subsets of 𝑈. Establish a formula for 𝑛(𝐴 ∪ 𝐵) in terms of 𝑛(𝐴), 𝑛(𝐵) and 𝑛(𝐴 ∩ 𝐵). Hence or otherwise deduce a formula for a particular case where A and B are disjoint?

Answer 1

We have that

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

because if we write $n(A)+n(B)$ we are counting each element of $A\cap B$ twice.

$A$ and $B$ are disjoint if they have no elements in common. That is, $n(A\cap B)=0$ . Therefore, substituting to the above formula we deduce that

$$n(A\cup B)=n(A)+n(B).$$