Question

List the members of these sets.

a) \(\left\{ {\left. x \right|x\; is a real number such that \;{x^2} = 1} \right\}\)

b) \(\left\{ {\left. x \right|x\; is a positive integer less than \;12} \right\}\)

c) \(\left\{ {\left. x \right|x\; is the square of an integer and \;x < 100} \right\}\)

d) \(\left\{ {\left. x \right|x\; is an integer such that \;{x^2} = 2} \right\}\)

Short answer

(a) The members of the given set are \(\left\{ { - 1,\;1} \right\}\).

(b) The members of the given set are \(\left\{ {1,{\rm{ }}2,3,4,5,6,7,8,9,10,11} \right\}\).

(c) The members of the given set are \(\left\{ {0,{\rm{ 1}},4,9,16,25,36,49,64,81} \right\}\).

(d) The member of the given set is \(\left\{ \emptyset \right\}\).

Step-by-step solution

Definitions of Set

An unordered collection of elements or members is known as set.

List the members of the set \(\left\{ {\left. x \right|x\;is a real number such that\;{x^2} = 1} \right\}\) (a)

Consider the given set is \(\left\{ {\left. x \right|x\;{\rm{is a real number such that}}\;{{\rm{x}}^2} = 1} \right\}\).

The square of \(1\) and \( - 1\) is \(1\). These are only numbers the square equal to \(1\).

Expect these two numbers, the square of any real number is either less than \(1\) or greater than \(1\).

Therefore, the members of the given set are \(\left\{ { - 1,\;1} \right\}\).

List the members of the set \(\left\{ {\left. x \right|x\;is a positive integer less than\;12} \right\}\)  (b)

Consider the given set.

The set of positive integers is \({Z^ + } = \left\{ {1,\;2,\;3,...} \right\}\).

The positive integers less than \(12\) are \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10\), and \(11\).

Therefore, the members of the given set are \(\left\{ {1,{\rm{ }}2,3,4,5,6,7,8,9,10,11} \right\}\).

List the members of the set\(\left\{ {\left. x \right|x\;is the square of an integer and\;x < 100} \right\}\) (c)

Consider the given set.

The members of the set consist perfect square of an integer and these numbers less than 100.

\(\begin{aligned}{c}{\left( { - 1} \right)^2} = {1^2}\\ = 1\\{\left( { - 2} \right)^2} = {2^2}\\ = 4\end{aligned}\)

And, for 3 and 4,

\(\begin{aligned}{c}{\left( { - 3} \right)^2} = {3^2}\\ = 9\\{\left( { - 4} \right)^2} = {4^2}\\ = 16\end{aligned}\)

And, for 5 and 6,

\(\begin{aligned}{c}{\left( { - 5} \right)^2} = {5^2}\\ = 25\\{\left( { - 6} \right)^2} = {6^2}\\ = 36\end{aligned}\)

And, for 7 and 8,

\(\begin{aligned}{c}{\left( { - 7} \right)^2} = {7^2}\\ = 49\\{\left( { - 8} \right)^2} = {8^2}\\ = 64\end{aligned}\)

And, for 9 and 10,

\(\begin{aligned}{c}{\left( { - 9} \right)^2} = {9^2}\\ = 81\\{\left( { - 10} \right)^2} = {10^2}\\ = 100\end{aligned}\)

Leave 10 and so on.

Therefore, the members of the given set are \(\left\{ {0,{\rm{ 1}},4,9,16,25,36,49,64,81} \right\}\).

List the members of the set \(\left\{ {\left. x \right|x\;is an integer such that\;{x^2} = 2} \right\}\)  (d)

Consider the given set.

It is known that 2 is not a perfect square of an integer.

Thus, there is no integer whose square is 2.

Therefore, the member of the given set is \(\left\{ \emptyset \right\}\).