Day 01- Discrete Mathematics for gate CSEIT- Concept of empty set or null set.

Day 01 – Discrete Mathematics for GATE CSE/IT

Concept of Empty Set (Null Set) 

 Definition:

An empty set (also called a null set) is a set that contains no elements. It is denoted by:
∅ (phi symbol)
{} (curly brackets with nothing inside)

 Example:

• The set of all prime numbers between 90 and 100: ∅ (since there are no prime numbers in this range).
• The set of all natural numbers less than zero: ∅ (as natural numbers start from 1).

 Important Properties of Empty Set

Subset Property: The empty set is a subset of every set.
Cardinality: The number of elements in an empty set is 0 → |∅| = 0.
Union Property: Any set A united with an empty set remains unchanged → A ∪ ∅ = A.
Intersection Property: Any set A intersected with an empty set gives an empty set → A ∩ ∅ = ∅.
Power Set: The power set of an empty set is P(∅) = {∅} (a set containing only the empty set).

 GATE Practice Question

Q1: What is the power set of the empty set?
(A) {∅}
(B) ∅
(C) {∅, {∅}}
(D) {{∅}}

Answer:  (A) {∅}

Trick: The power set contains all subsets of a set. Since ∅ has no elements, its only subset is itself!

 Summary:

The empty set (∅) is a fundamental concept in Discrete Mathematics and is used in set theory, logic, and algorithms. It plays a key role in solving problems related to relations, functions, and probability in GATE CSE/IT.

Day 01- Discrete Mathematics for gate CSEIT- Concept of empty set or null set.

Discrete Mathematics for Computer Science

UNIT – I – Set Theory – SMT5201

Empty set

Great! Let’s break down the concept of the empty set (null set) from Day 01 of Discrete Mathematics for GATE CSE/IT in a simple, beginner-friendly way.

Day 01 – Discrete Mathematics for GATE CSE/IT

Topic: Concept of Empty Set (Null Set)

What is an Empty Set?

An empty set (also called null set) is a set that has no elements.

Definition:

“A set which does not contain any element is called an empty set or null set.”

Symbol:

• It is denoted by:
  $\varnothing \text{or}\{\}$

Examples of Empty Sets

1. Set of prime numbers less than 2 → $\varnothing$

2. Set of natural numbers less than 0 → $\varnothing$

3. Set of students in a class who scored more than 100 out of 100 → $\varnothing$

Important Notes for GATE:

1. Cardinality of Empty Set:

• Cardinality = number of elements = 0
  So, $|\varnothing |=0$

2. Subset Property:

• The empty set is a subset of every set
  $\varnothing \subseteq A\text{for any set }A$

3. Power Set of Empty Set:

• Power set = Set of all subsets

• Power set of $\varnothing$:
  $P(\varnothing )=\{\varnothing \}$
  So, cardinality of power set = 1

Common GATE MCQ Concept:

Q. What is the power set of the empty set?
Answer: $\{\varnothing \}$

Q. Is empty set a subset of itself?
Answer: Yes

Q. What is the cardinality of $P(P(\varnothing ))$?

• First, $P(\varnothing )=\{\varnothing \}$

• Then, $P(P(\varnothing ))=\{\varnothing ,\{\varnothing \}\}$
  So, cardinality = 2

Easy Trick to Remember:

❝ Think of the empty set as an empty box — it’s still a box (a valid set), but it holds nothing inside. ❞

Summary:

Want More?

I can send you:

• PDF notes of Set Theory for GATE

• Practice GATE-level MCQs on Sets

• Day-by-day Discrete Math plan for GATE

Would you like that?

Day 01- Discrete Mathematics for gate CSEIT- Concept of empty set or null set.

Computer Science & Engineering Syllabus

Department of Computer Science & Engineering