Chapter 1: Sets | SJMaths Class 11

1. What is a Set?

A set is a well-defined collection of distinct objects. "Well-defined" means that for any object, we can clearly say whether it belongs to the collection or not.

Well-Defined?

"Collection of all vowels in English alphabet"

(Click to reveal)

YES

We know exactly what they are: {a, e, i, o, u}. This IS a set.

Well-Defined?

"Collection of best cricket players"

(Click to reveal)

NO

"Best" is subjective. My list might differ from yours. This is NOT a set.

2. Representation of Sets

Form	Description	Example
Roster (Tabular)	Elements listed within braces { }, separated by commas.	\( A = \{2, 4, 6, 8\} \)
Set-Builder	Describes the property possessed by all elements.	\( A = \{x : x \text{ is an even natural number } < 10 \} \)

3. Types of Sets

Empty Set (\( \phi \)): A set with no elements. Ex: \( \{x : x^2 = 4, x \text{ is odd}\} \).
Singleton Set: A set with exactly one element. Ex: \( \{5\} \).
Finite Set: Countable number of elements. Ex: Days in a week.
Infinite Set: Uncountable elements. Ex: Natural numbers \( \mathbb{N} \).
Equal Sets: Two sets having exactly the same elements. Order doesn't matter.

3.1 Universal Set & Complement

The Universal Set (U) is the set that contains all elements under consideration.

The Complement of a set A, denoted by \( A' \), is the set of all elements of U which are not in A.

Example:

If \( U = \{1,2,3,4,5\} \) and \( A = \{1,3\} \), then
\( A' = \{2,4,5\} \)

3.2 Disjoint Sets

Two sets A and B are called disjoint if they have no common element.

\( A \cap B = \phi \)

Example:

\( A = \{1,3,5\}, \; B = \{2,4,6\} \)

4. Subsets & Power Sets

Set A is a subset of B (\( A \subset B \)) if every element of A is also in B.

Power Set \( P(A) \): The collection of all subsets of A. If \( n(A) = m \), then \( n(P(A)) = 2^m \).

Power Set Generator

Enter elements separated by commas (max 3 for demo).

5. Venn Diagrams & Operations

Union (\( \cup \)): Elements in A OR B.
Intersection (\( \cap \)): Elements in BOTH A AND B.
Difference (\( A-B \)): Elements in A but NOT in B.

Venn Explorer

Set Operations Lab

Enter elements for Set A and Set B.

A ∪ B:

A ∩ B:

A - B:

B - A:

5.1 Laws of Set Operations

Law	Expression
Commutative	\( A \cup B = B \cup A \) \( A \cap B = B \cap A \)
Associative	\( A \cup (B \cup C) = (A \cup B) \cup C \)
Distributive	\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
Identity	\( A \cup \phi = A \), \( A \cap U = A \)

5.2 De Morgan’s Laws

These laws relate union, intersection, and complements.

\( (A \cup B)' = A' \cap B' \)

\( (A \cap B)' = A' \cup B' \)

⚠️ CBSE frequently asks verification using Venn diagrams.

6. Practical Problems formula

For finite sets A and B:

\( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)

6.1 Problems Involving Three Sets

When three sets A, B, and C are involved:

\( n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A\cap B)-n(B\cap C)-n(A\cap C) + n(A\cap B\cap C) \)

✔ Use Venn diagram before substituting values

Standard Sets & Symbols

\( \mathbb{N} \): Natural Numbers
\( \mathbb{Z} \): Integers
\( \mathbb{Q} \): Rational Numbers
\( \mathbb{R} \): Real Numbers
\( \phi \): Empty Set

Common Mistakes Students Make ❌

❌ Treating subjective collections as sets
❌ Writing repeated elements in roster form
❌ Forgetting complement is with respect to U
❌ Applying De Morgan’s laws incorrectly

CBSE Exam Question Patterns 🎯

Find number of subsets
Verify De Morgan’s laws
Solve word problems using Venn diagrams
Find missing values using set formula
Represent data using Venn diagram

Concept Mastery Quiz

1. If \( A = \{1, 2\} \), how many subsets does it have?

2. \( A \cup A' \) is equal to:

3. Which set is infinite?

4. If \( A \subset B \), then \( A \cap B \) is:

5. The set of "intelligent students" is: