Class 11 | CBSE | MATHEMATICS
MATHEMATICS | INTERVALS OF SETS

Sudying Set with a context

In mathematics, we never study sets in isolation. We always study them within a context.

The brain understands faster when it knows: “What is the whole picture?”

What is a Universal Set?

In a given situation, the set that contains all objects under discussion is called the Universal Set.

It is usually denoted by the letter U.

( Brain-Friendly ) Definition

Universal Set = The biggest possible set in a given context.

All other sets we talk about are subsets of this universal set.

Notation and Symbols

Universal Set → U
Its subsets → A, B, C, …

Visualise:

• U as a big box
• A, B, C as smaller boxes inside it

📌 Example 1: Number System

While studying numbers, we may focus on:

• Natural numbers
• Prime numbers
• Even numbers

All these sets exist inside a larger set.

That larger set is the Universal Set.

📌 Example:

If we are studying natural numbers, then:

U = N
Prime numbers ⊂ U
Even numbers ⊂ U

📌 Example 2: Integers and Rational Numbers

If our basic discussion is about integers, then:

The universal set can be:

• The set of rational numbers (Q)
• Or the set of real numbers (ℝ)

Key Insight:
A universal set is not fixed. It depends on the context of discussion.

📌 Example 3: Real-Life Context

In human population studies:

The universal set consists of all people in the world.

Subsets may include:

• People of a particular country
• People of a certain age group
• Students, teachers, professionals, etc.

Very Important Observations (Exam Focus)

• Universal set depends on the situation
• All sets under discussion must be subsets of U
• The same set can be universal in one case and a subset in another

Final Brain Snapshot

• Universal Set = complete collection in a context
• Denoted by U
• Other sets are subsets of U
• Context decides the universal set