Question

Determine whether each of these pairs of sets are equal.

(a) \(\left\{ {1,\;3,\;3,\;3,\;5,\;5,\;5,\;5,\;5} \right\},\;\left\{ {5,\;3,\;1} \right\}\)

(b) \(\left\{ {\left\{ 1 \right\}} \right\},\left\{ {1,\left\{ 1 \right\}} \right\}\)

(c) \(\emptyset ,\left\{ \emptyset \right\}\)

Short answer

(a) The sets are equal.

(b) The sets are not equal.

(c) The sets are not equal..

Step-by-step solution

Definitions of Equal set

If the elements of both sets are common irrespective of repetition or the order then the sets said to be equal.

Determine whether sets \(\left\{ {1,\;3,\;3,\;3,\;5,\;5,\;5,\;5,\;5} \right\},\;\left\{ {5,\;3,\;1} \right\}\) are equal. (a)

In the given sets, look only at the distinct elements, not the repetition or the order. The given sets contain common elements \(1,\;3\), and \(5\). So, they are equal.

Therefore, the sets are equal.

Determine whether sets \(\left\{ {\left\{ 1 \right\}} \right\},\left\{ {1,\left\{ 1 \right\}} \right\}\) are equal. (b)

In the given sets, the first set is subset of the second set. The first set consists an element which is a set containing \(1\) that is \(\left\{ 1 \right\}\), whereas, the second set has two distinct elements \(1\) and \(\left\{ 1 \right\}\), so they cannot be equal.

Therefore, the sets are not equal.

Determine whether sets \(\emptyset ,\left\{ \emptyset  \right\}\) are equal. (c)

The first set is an empty set, whereas the second set is a set containing an empty set. So, they are distinct.

Therefore, the sets are not equal.