Question

Show that if \(A\) and \(B\) are finite sets, then \(A \cup B\) is a finite set.

Short answer

Some elements may be repeated which will only lower the number of elements in \(A \cup B\), where A and B are finite sets.

Step-by-step solution

Step 1

Since \(A\) is a finite set, we can enumerate its elements as \({a_1},{a_2},.....,{a_n}\) for some positive integer \(n\).

Since \(B\) is a finite set, we can enumerate its elements as \({b_1},{b_2},.....,{b_n}\) for some positive integer \(m\).

Step 2

We can label the elements in the set using whatever indices.

Thus, relabel the elements of \(B\)so that the index starts from \(n + 1\)

Therefore, the elements are now listed as \({b_{n + 1}},{b_{n + 2}},.....,{b_{n + m}}\)

\(A \cup B\)has the elements \({a_1},{a_2},.....,{a_n},{b_{n + 1}},{b_{n + 2}},.....,{b_{n + m}}\)which is almost \(n + m\)elements

Hence, some elements may be repeated which will only lower the number of elements in \(A \cup B\)