Question

Let X = {0, 1, 2}. Consider the power set P(X). Recall that P(X) = {A such that A ⊆ X}.
Determine if each statement is true or false. (a) X ∈ P(X).
(b) ∅∈P(X)
(c) X ⊆ P(X).
(d) {0,2}∈P(X) (e) {0,2,1} ∈ X (f) {0}⊆X
(g) {{0}} ⊆ P(X)

Answer 1

Answer on Question #76365 – Math – Discrete Mathematics

Question

Let $X = \{0,1,2\}$. Consider the power set $P(X)$. Recall that $P(X) = \{A \text{ such that } A \subseteq X\}$. Determine, if each statement is true or false.

(a) $X \in P(X)$;

(b) $\varnothing \in P(X)$;

(c) $X \subseteq P(X)$;

(d) $\{0,2\} \in P(X)$;

(e) $\{0,2,1\} \in X$;

(f) $\{0\} \subseteq X$;

(g) $\{\{0\}\} \subseteq P(X)$.

Solution

(a). Since $X \subseteq X$ for the set $X$, then $X \in P(X)$ by definition of $P(X)$.

Answer: The statement is true.

(b). Since $\varnothing \subseteq X$ for $X$, then $X \in P(X)$ by definition of $P(X)$.

Answer: The statement is true.

(c). The set $P(X)$ consists of all subsets of $X$. Consequently, every subset of $X$ lies in $P(X)$ as an element, but not as a subset. Therefore the statement $X \subseteq P(X)$ is not true.

Answer: The statement is false.

(d). Since $\{0,2\} \subseteq X$, then $\{0,2\} \in P(X)$.

Answer: The statement is true.

(e). The set $X$ consists of elements $0, 1, 2$ only. Consequently, $\{2, 0, 1\} \notin X$.

Answer: The statement is false.

(f). Since $0 \in X$, then the set $\{0\}$ (which consists of only $\emptyset$) is a subset of $X$, i.e. $\{0\} \subseteq X$.

Answer: The statement is true.

(g). Since $\{0\} \in P(X)$, then the set $\{\{0\}\}$ (which consist of only $\{0\}$ element of $P(X)$) is a subset of $P(X)$, i.e. $\{\{0\}\} \subseteq P(X)$.

Answer: The statement is true.

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