Question

Show that if A and B are finite sets, then $|A\cap B|\le |A\cup B|$. Determine when this relationship is an equality.

Short answer

$|A\cap B|\le |A\cup B|$

Step-by-step solution

Step: 1

Every element in the sets A and B should be also in $A\cup B$.

So,

$$\begin{gathered} A\subseteq A\cup B \\ B\subseteq A\cup B \end{gathered}$$

Step: 2

Similarly,

$$\begin{gathered} A\cap B\subseteq A \\ A\cap B\subseteq B \end{gathered}$$

We get$A\cap B\subseteq A\cup B$

Step: 3

Therefore, the cardinality of these two sets$|A\cap B|\le |A\cup B|$