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CBSE • Class XI • Mathematics • Ch 1
Estimated Time: 45 Mins
Study Progress: In Progress

Sets

In Class 11 Mathematics, "Sets" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

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📦 Have You Ever Wondered?

How does a search engine instantaneously find results matching 'colleges AND scholarships NOT loans', or how do mathematicians build the entire univer...

How does a search engine instantaneously find results matching 'colleges AND scholarships NOT loans', or how do mathematicians build the entire universe of numbers from an empty bag? Set theory is the axiomatic bedrock of all modern mathematics.

Why This Chapter Matters

In Class 11 Mathematics, "Sets" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Venn diagrams and sets introduction.
  • Natural numbers and integers.
  • Logical operations.

What You Will Learn (Core Objectives)

  • Represent sets using Roster (Tabular) and Set-Builder notation.
  • Distinguish between Empty, Finite, Infinite, and Equal sets.
  • Define Subsets, Power Sets ($P(A)$ with $2^n$ elements), and Universal Sets.
  • Perform Set Operations: Union ($A \cup B$), Intersection ($A \cap B$), and Difference ($A - B$).
  • State and prove De Morgan's Laws: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$ using Venn diagrams.

Chapter Roadmap & Progression

1 1. Set Representation & Types
2 2. Operations on Sets & Venn Diagra...
3 3. De Morgan's Laws & Cardinality

Complete Concept Guide (100% Curriculum Coverage)

1. Set Representation & Types

A Set is a well-defined collection of distinct objects:
• Roster form: $V = \{a, e, i, o, u\}$.
• Set-Builder form: $A = \{x : x \in \mathbb{N} \text{ and } 1 < x < 6\}$.
• Power Set $P(A)$: The set of all subsets of $A$. If $|A| = n$, then: $$\mathbf{|P(A)| = 2^n}$$

2. Operations on Sets & Venn Diagrams

  • Union ($A \cup B$): Elements belonging to $A$, $B$, or both ($x \in A \text{ or } x \in B$).
  • Intersection ($A \cap B$): Elements common to both ($x \in A \text{ and } x \in B$). If $A \cap B = \emptyset$, they are Disjoint sets.
  • Difference ($A - B$): Elements belonging to $A$ but NOT to $B$.
  • Complement ($A'$): Elements in Universal set $U$ not in $A$: $A' = U - A$.

3. De Morgan's Laws & Cardinality

De Morgan's Laws govern set duality: $$\mathbf{(A \cup B)' = A' \cap B'} \quad \text{and} \quad \mathbf{(A \cap B)' = A' \cup B'}$$ For two finite sets: $$\mathbf{n(A \cup B) = n(A) + n(B) - n(A \cap B)}$$

Visual Learning & Conceptual Map

Sets Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Set Representation & Types • 2. Operations on Sets & Venn Diagrams

Chapter Summary & 10 Key Takeaways

Takeaway 1
Well-Defined: Unambiguous membership criteria where an element definitely belongs or does not.
Takeaway 2
Power Set: The set of all subsets, always containing $\emptyset$ and the set itself with $2^n$ elements.
Takeaway 3
Disjoint Sets: Two sets whose intersection is strictly the empty set ($A \cap B = \emptyset$).
Takeaway 4
De Morgan's Laws: Duality transforming complement of union into intersection of complements.
Takeaway 5
Venn Diagram: Geometric visualization mapping set relationships using circles inside a rectangle.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
If set $A$ has 3 elements, how many subsets and how many proper subsets does $A$ have?
Reveal Answer & Explanation
Answer: Total subsets $= 2^n = 2^3 = 8$. Proper subsets $= 2^n - 1 = 8 - 1 = 7$.
8 subsets, 7 proper subsets.
2
State De Morgan's Laws for any two sets $A$ and $B$.
Reveal Answer & Explanation
Answer: (1) $(A \cup B)' = A' \cap B'$, (2) $(A \cap B)' = A' \cup B'$.
Complement of union is intersection of complements.
3
If $X$ and $Y$ are two sets such that $X \cup Y$ has 50 elements, $X$ has 28 elements and $Y$ has 32 elements, find $n(X \cap Y)$.
Reveal Answer & Explanation
Answer: $n(X \cap Y) = n(X) + n(Y) - n(X \cup Y) = 28 + 32 - 50 = 60 - 50 = 10$.
n(X ∩ Y) = 10.
4
Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $A - B$ and $B - A$.
Reveal Answer & Explanation
Answer: $A - B = \{1, 2\}$ and $B - A = \{5, 6\}$.
A - B = {1, 2}; B - A = {5, 6}.
5
What is the power set of the empty set $\emptyset$?
Reveal Answer & Explanation
Answer: $P(\emptyset) = \{\emptyset\}$, which has $2^0 = 1$ element.
P(∅) = {∅} with 1 element.
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Timed CBT Practice Tests (Exam Simulator)

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