Question

Determine whether each of these statements is true or false.

a) \(x \in \left\{ x \right\}\)

b) \(\left\{ x \right\} \subseteq \left\{ x \right\}\)

c) \(\left\{ x \right\} \in \left\{ x \right\}\)

d) \(\left\{ x \right\} \in \left\{ {\left\{ x \right\}} \right\}\)

e) \(\emptyset \subseteq \left\{ x \right\}\)

f) \(\emptyset \in \left\{ x \right\}\)

Short answer

(a) True

(b) True

(c) False

(d) True

(e) True

(f) False

Step-by-step solution

Definitions of \(\emptyset \), and Subset

The symbol\(\emptyset \)represents the empty set and the empty set does not contain any elements.

Xis a subset of Y if every element of X is also an element of Y. It is represented as\(X \subset Y\).

Determine whether the statement \(x \in \left\{ x \right\}\) is true or false (a)

Consider the given statement.

\(x \in \left\{ x \right\}\)

The meaning of the given statement is that \(x\) is an element of the set \(\left\{ x \right\}\). The set \(\left\{ x \right\}\) contains only the element of \(x\).

Hence, the given statement is true.

Determine whether the statement \(\left\{ x \right\} \subseteq \left\{ x \right\}\) is true or false (b)

Consider the given statement.

\(\left\{ x \right\} \subseteq \left\{ x \right\}\)

The meaning of the given statement is that the set \(\left\{ x \right\}\) is an inclusive subset of \(\left\{ x \right\}\). Every subset is an inclusive subset of itself.

Therefore, the given statement is true.

Determine whether the statement \(\left\{ x \right\} \in \left\{ x \right\}\) is true or false (c)

Consider the given statement.

\(\left\{ x \right\} \in \left\{ x \right\}\)

The meaning of the given statement is that the set \(\left\{ x \right\}\) is an element of the set \(\left\{ x \right\}\). The given set \(\left\{ x \right\}\) does not contain any sets as elements.

Therefore, the given statement is false.

Determine whether the statement \(\left\{ x \right\} \in \left\{ {\left\{ x \right\}} \right\}\) is true or false (d)

Consider the given statement.

\(\left\{ x \right\} \in \left\{ {\left\{ x \right\}} \right\}\)

The meaning of the given statement is that the \(\left\{ x \right\}\) is an element of the set \(\left\{ {\left\{ x \right\}} \right\}\). The set \(\left\{ {\left\{ x \right\}} \right\}\) contains only the element \(\left\{ x \right\}\).

Therefore, the given statement is true.

Determine whether the statement \(\emptyset  \subseteq \left\{ x \right\}\) is true or false (e)

Consider the given statement.

\(\emptyset \subseteq \left\{ x \right\}\)

The meaning of the given statement is that the set \(\emptyset \) is an inclusive subset of the set \(\left\{ x \right\}\). The empty set is a subset of every set.

Therefore, the given statement is true.

Determine whether the statement \(\emptyset  \in \left\{ x \right\}\) is true or false (f)

Consider the given statement.

\(\emptyset \in \left\{ x \right\}\)

The meaning of the given statement is that the set \(\emptyset \) is an element of the set \(\left\{ x \right\}\). The set \(\left\{ x \right\}\) contains only the element \(x\) and thus, it does not contain the element \(\emptyset \).

Therefore, the given statement is false.