What is Set Theory?

Set theory is the part of mathematics that deals with collections of objects, known as sets. These objects can be numbers, letters, people, or even ideas — anything that can be grouped meaningfully.

A set is just a group of clearly defined and distinct objects. For example, if we say "set of vowels in English", it clearly means:

{a, e, i, o, u}

This field of math was developed and formalized by Georg Cantor, who is considered the founder of modern set theory.

Real-Life Examples of Sets:

• The list of students enrolled in your class
• Planets in the solar system
• Books kept in a specific rack in the library

These are all examples of sets because they group similar items under one label.

Set Notation:

Sets are usually named with capital letters like A, B, C... And the elements inside the set are written in curly brackets.

Example: Let A = {2, 4, 6, 8}

If 2 is in set A, we write it as: 2 ∈ A (which means 2 belongs to A)

If 5 is not in A, we write: 5 ∉ A

Key Terms You Should Know:

Some Important Points:

• The order of elements doesn’t matter in a set. Example: {1, 2, 3} is the same as {3, 2, 1}
• Repetition is not allowed. So {1, 2, 2, 3} is just written as {1, 2, 3}

Source/Reference:

This content is based on NCERT Class 11 – Chapter 1: Sets (Pages 1–3) and aligned with Christ University BSc DS-AI course plan (Unit 1: Set Theory).

1. Roster Form (Tabular Form)

In this method, all elements of the set are listed inside curly brackets.

Example: Let A be the set of even numbers less than 10.

A = {2, 4, 6, 8}

Things to note:

• Elements are separated by commas.
• Order does not matter: {2, 4, 6} = {6, 2, 4}
• No element is repeated: {1, 2, 2, 3} → {1, 2, 3}

2. Set-Builder Form

Instead of listing elements, this form describes a rule or property that elements of the set follow.

Example:

A = {x : x is an even number less than 10}

We read this as: A is the set of all x such that x is an even number less than 10.

Sometimes written as: A = {x ∈ N : x < 10 and x is even}

Examples to Understand Clearly:

Important Symbols to Remember:

1. Empty Set (Null Set)

A set with no elements. Symbol: ∅ or { }

Example: Let A = {x : x is a natural number less than 1} → A = ∅

2. Singleton Set

A set with exactly one element.

Example: B = {7} is a singleton set.

3. Finite Set

A set that contains a countable number of elements.

Example: C = {2, 4, 6, 8} → has 4 elements.

4. Infinite Set

A set with endless elements, not countable.

Example: D = {x : x is a natural number} → D = {1, 2, 3, 4, ...}

5. Equal Sets

Two sets are equal if they contain exactly the same elements, irrespective of order.

Example: E = {1, 2, 3}, F = {3, 1, 2} → E = F

6. Equivalent Sets

Two sets with the same number of elements, not necessarily the same elements.

Example: G = {a, b, c}, H = {1, 2, 3} → Equivalent sets (same cardinality)

7. Subset

A set A is a subset of B if every element of A is also in B. Notation: A ⊆ B

Example: A = {1, 2}, B = {1, 2, 3} → A ⊆ B

8. Proper Subset

A ⊂ B means A is a subset of B but not equal to B.

9. Universal Set

The set that contains all the elements under consideration.

Example: In class 11 sets, U = set of all natural numbers (N)

10. Power Set

The set of all subsets of a given set.

Example: A = {1, 2} → P(A) = {∅, {1}, {2}, {1, 2}}

11. Disjoint Sets

Two sets with no common elements. A ∩ B = ∅

Example: A = {1, 2}, B = {3, 4} → disjoint sets

Common Operations:

• Union (A ∪ B): All elements from A or B or both.
• Intersection (A ∩ B): Elements common to both A and B.
• Difference (A – B): Elements in A but not in B.
• Complement (A′): All elements not in A, but in Universal set.

Example:

Let U = {1, 2, 3, 4, 5, 6}, A = {2, 4, 6}, B = {1, 2, 3}

Then:

• A ∪ B = {1, 2, 3, 4, 6}
• A ∩ B = {2}
• A – B = {4, 6}
• B′ = {4, 5, 6}

References:

• NCERT Class 11 Mathematics — Chapter 1: Sets (Pages 5–7)
• ML Aggarwal Class 11 – Set Theory, Chapter 5, Intro Pages

1. Idempotent Laws

• A ∪ A = A
• A ∩ A = A

2. Domination Laws

• A ∪ U = U
• A ∩ ∅ = ∅

3. Identity Laws

• A ∪ ∅ = A
• A ∩ U = A

4. Complement Laws

• A ∪ A′ = U
• A ∩ A′ = ∅

5. Double Complement Law

• (A′)′ = A

6. Commutative Laws

• A ∪ B = B ∪ A
• A ∩ B = B ∩ A

7. Associative Laws

• (A ∪ B) ∪ C = A ∪ (B ∪ C)
• (A ∩ B) ∩ C = A ∩ (B ∩ C)

8. Distributive Laws

• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

References Used:

• NCERT Class 11 – Chapter 1: Sets, Pages 7–9
• ML Aggarwal Class 11 – Chapter 5: Set Theory, Pages 35–37

Let:

• A = set of students who play football
• B = set of students who play cricket
• A ∩ B = those who play both

Use formula:

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Example:

• 30 play football
• 25 play cricket
• 10 play both

How many play either?

n(A ∪ B) = 30 + 25 – 10 = 45 students

Short Answer:

• Define: Set, Subset, Universal Set.
• State and prove: A ∪ A' = U
• Draw Venn diagram for:
  • A ∩ (B ∪ C)
  • (A ∩ B)' ∪ C

Application-Based:

In a group of 100 students:

• 60 like Tea, 45 like Coffee, 20 like both

How many like only Tea?

How many like either?

How many like neither?

Prove using laws:

(A ∩ B) ∪ A = A

Proof:

(A ∩ B) ∪ A
= A ∪ (A ∩ B)  (Commutative Law)
= A             (Absorption Law)

(Verified) Q.E.D.

* References for All Chapters:

• NCERT Class 11 Mathematics — Chapter 1: Sets (Pages 1–11)
• ML Aggarwal Class 11 ISC Mathematics — Chapter 5: Set Theory (Pages 1–39)
• Christ University Syllabus — Unit 1: Foundation of Mathematics