GATE CS 2015 Set 1 Q.26 (CS Q.16): Sets, Relations, Functions, Partial Orders and Lattices

For a set $A$ , the power set of $A$ is denoted by $2^A$ . If $A = \{5, \{6\}, \{7\}\}$ , which of the following options are TRUE?

I. $\emptyset \in 2^A$
II. $\emptyset \subseteq 2^A$
III. $\{5, \{6\}\} \in 2^A$
IV. $\{5, \{6\}\} \subseteq 2^A$

I and II only

II and III only

I, II and III only

I and IV only

Official answer and solution

Solution

Recall the definitions of set membership ( $\in$ ) and subset inclusion ( $\subseteq$ ) with respect to power sets.

Given $A = \{5, \{6\}, \{7\}\}$ . The power set $2^A$ is the set of all subsets of $A$ .

Evaluating each proposition:

$\emptyset \in 2^A$ : Since the empty set is a subset of every set, $\emptyset \subseteq A$ . By definition of power set, $\emptyset \in 2^A$ . (True)
$\emptyset \subseteq 2^A$ : The empty set is a subset of every set, including $2^A$ . (True)
$\{5, \{6\}\} \in 2^A$ : The elements $5$ and $\{6\}$ both belong to $A$ . Therefore, the set $\{5, \{6\}\}$ is a subset of $A$ , meaning $\{5, \{6\}\} \in 2^A$ . (True)
$\{5, \{6\}\} \subseteq 2^A$ : For $\{5, \{6\}\} \subseteq 2^A$ to hold, every element in $\{5, \{6\}\}$ must be an element of $2^A$ . This would require $5 \in 2^A$ (meaning $5 \subseteq A$ ), but $5$ is an integer (an atom), not a set. Hence, $\{5, \{6\}\} \not\subseteq 2^A$ . (False)

Therefore, I, II and III are true (option C).

Final answer

Common trap

Confusing membership $\in$ with subset inclusion $\subseteq$ for set elements in statement IV.

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