Don't know how to solve a question regarding sets

Question

The next definition is giving in the book the exercise was proposed on.

The definition reads:

Let $a=n(A)$ and $b=n(B)$ where $A$ and $B$ are two disjoint finite sets. Then $a+b=n(A∪B).$

I understand the fact of A and B being disjoint is a must. Yet, what should I do in case A and B have at least an element in common?

I was proposed an exercise in which I am to compute the cardinal of two non-disjoint sets given. But the definition on my mathbook does not cover non-disjoint sets. What am I supposed to do?

It depends on how the sets are given... but you can use $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. — Jair Taylor, Dec 13 '20 at 21:31
Yup, try it out yourself on some examples and you'll see that it works. — Jair Taylor, Dec 13 '20 at 21:35

score 2 · Accepted Answer · answered Dec 13 '20 at 21:32

The reason the formula $|A \cup B| = |A| + |B|$ is only applicable to the case when $A$ and $B$ are disjoint is that it would double-count the elements in the intersection $A \cap B$. Thus, it can be generalized to the case of not necessarily disjoint sets as

$$ |A \cup B| = |A| + |B| - |A \cap B|\tag1. $$

This can be seen by considering the three possible cases for $x \in A \cup B$:

$x \in A$ and $x \notin B$
$x \notin A$ and $x \in B$
$x \in A$ and $x \in B$

and verifying that $x$ is counted exactly once in $(1)$.

This reasoning is called the inclusion-exclusion principle and can be generalized to more than two sets.

Don't know how to solve a question regarding sets

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