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

Short Answer

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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

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01

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 \(m\).

02

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 \(A \cup B\)

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