Question

What is the cardinality of each of these sets?

(a) \(\left\{ {\bf{a}} \right\}\)

(b) \(\;\left\{ {\left\{ {\bf{a}} \right\}} \right\}\)

(c) \(\left\{ {{\bf{a}}\left\{ {\bf{a}} \right\}} \right\}\)

(d) \(\left\{ {{\bf{a}}\left\{ {\bf{a}} \right\}{\bf{,}}\left\{ {{\bf{a}}\left\{ {\bf{a}} \right\}} \right\}} \right\}\)

Short answer

(a) \(\left\{ a \right\}\)has cardinality of 1.

(b) \(\left\{ {\left\{ a \right\}} \right\}\)has a cardinality of 1.

(c) \(\left\{ {a,\left\{ a \right\}} \right\}\)has cardinality of 2.

(d) \(\left\{ {a,\left\{ a \right\},\left\{ {a,\left\{ a \right\}} \right\}} \right\}\)has cardinality of 3.

Step-by-step solution

Cardinality of sets

The measure of the number of elements of the set is the cardinality of a set.

Therefore, it denotes the size of a finite set.

To determine the cardinality of given set (a)

The cardinality of \(\left\{ a \right\}\) is equals to 1 since it contains only 1 element.

Thus, the given pair of set \(\left\{ a \right\}\)has a cardinality of 1.

To determine the cardinality of given set (b)

The cardinality of \(\left\{ {\left\{ a \right\}} \right\}\) is equals to 1 since it contains only 1 element.

Thus, the given pair of set \(\left\{ {\left\{ a \right\}} \right\}\)has a cardinality of 1.

To determine the cardinality of given set (c)

The cardinality of \(\left\{ {a,\left\{ a \right\}} \right\}\) is equals to 2 since it contains 2 elements.

Thus, the given pair of set \(\left\{ {a,\left\{ a \right\}} \right\}\)has a cardinality of 2.

To determine the cardinality of given set (d)

The cardinality of \(\left\{ {a,\left\{ a \right\},\left\{ {a,\left\{ a \right\}} \right\}} \right\}\) is equals to 3 since it contains 3 elements.

Thus, the given pair of set \(\left\{ {a,\left\{ a \right\},\left\{ {a,\left\{ a \right\}} \right\}} \right\}\)has a cardinality of 3.