GATE CS 2015 Set 1 Q1 — Sets, Relations and Functions

Given the set $A = \{5, \{6\}, \{7\}\}$. The elements of $A$ are:

• $5$
• $\{6\}$
• $\{7\}$

The power set $2^A$ is the set of all subsets of $A$:

$$2^A = \Big\{ \varnothing, \{5\}, \{\{6\}\}, \{\{7\}\}, \{5, \{6\}\}, \{5, \{7\}\}, \{\{6\}, \{7\}\}, \{5, \{6\}, \{7\}\} \Big\}$$

Let's evaluate each statement:

1. $\varnothing \in 2^A$: TRUE. Since the empty set $\varnothing$ is a subset of $A$, it is an element of the power set $2^A$.
2. $\varnothing \subseteq 2^A$: TRUE. The empty set $\varnothing$ is a subset of every set, including $2^A$.
3. $\{5, \{6\}\} \in 2^A$: TRUE. Since $5 \in A$ and $\{6\} \in A$, the set $\{5, \{6\}\}$ is a subset of $A$, which makes it an element of $2^A$.
4. $\{5, \{6\}\} \subseteq 2^A$: FALSE. For this to be true, the elements $5$ and $\{6\}$ must belong to $2^A$. However, $5 \notin 2^A$ (only $\{5\} \in 2^A$) and $\{6\} \notin 2^A$ (only $\{\{6\}\} \in 2^A$).

Therefore, statements I, II, and III are true.