The GATE Grind

GATE 2015 CS – Question 20

Engineering Mathematics · Discrete Mathematics: Sets, Relations, Functions, Partial Orders and Lattices · 1 mark · Multiple choice

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. $\{5, \{6\}\} \subseteq 2^A$

  1. I and III only
  2. II and III only
  3. I, II and III only
  4. I, II and IV only

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) I, II and III only

Explanation

$2^A$ is the set of all subsets of $A$. The empty set is a subset of $A$, so $\emptyset \in 2^A$ (I). The empty set is a subset of every set, so $\emptyset \subseteq 2^A$ (II). $\{5, \{6\}\}$ is a subset of $A$, so it is an element of $2^A$ (III). For IV, the element 5 would have to be a subset of $A$, but 5 is not a set, so IV is false.