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CBSE • कक्षा XI • Mathematics • अध्याय 1

अनुमानित समय: 45 Mins

प्रगति: अध्ययनरत

समुच्चय (Sets)

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

अध्याय CBT ऑनलाइन टेस्ट दें सभी अध्याय देखें

प्रस्तावना अध्याय अवधारणाएँ अध्याय सारांश स्व-मूल्यांकन अभ्यास टेस्ट दें

📦 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.

यह अध्याय क्यों महत्वपूर्ण है

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

अध्ययन से पूर्व (आवश्यक ज्ञान)

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

इस अध्याय के लक्ष्य

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.

अध्याय रूपरेखा एवं प्रगति

1 1. Set Representation & Types

2 2. Operations on Sets & Venn Diagra...

3 3. De Morgan's Laws & Cardinality

सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन

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: $$\mathbf{n(A \cup B) = n(A) + n(B) - n(A \cap B)}$$

चित्रात्मक व्याख्या एवं मॉडल

Sets Master Matrix

Conceptual framework, core mechanisms, and analytical relationships

Academic Architecture

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

अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष

मुख्य बिंदु 1

Well-Defined: Unambiguous membership criteria where an element definitely belongs or does not.

मुख्य बिंदु 2

Power Set: The set of all subsets, always containing $\emptyset$ and the set itself with $2^n$ elements.

मुख्य बिंदु 3

Disjoint Sets: Two sets whose intersection is strictly the empty set ($A \cap B = \emptyset$).

मुख्य बिंदु 4

De Morgan's Laws: Duality transforming complement of union into intersection of complements.

मुख्य बिंदु 5

Venn Diagram: Geometric visualization mapping set relationships using circles inside a rectangle.

स्व-मूल्यांकन अभ्यास (Check Your Understanding)

मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।

If set $A$ has 3 elements, how many subsets and how many proper subsets does $A$ have?

उत्तर एवं व्याख्या देखें

उत्तर: Total subsets $= 2^n = 2^3 = 8$. Proper subsets $= 2^n - 1 = 8 - 1 = 7$.
8 subsets, 7 proper subsets.

State De Morgan's Laws for any two sets $A$ and $B$.

उत्तर एवं व्याख्या देखें

उत्तर: (1) $(A \cup B)' = A' \cap B'$, (2) $(A \cap B)' = A' \cup B'$.
Complement of union is intersection of complements.

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 $n(X \cap Y)$.

उत्तर एवं व्याख्या देखें

उत्तर: $n(X \cap Y) = n(X) + n(Y) - n(X \cup Y) = 28 + 32 - 50 = 60 - 50 = 10$.
n(X ∩ Y) = 10.

Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $A - B$ and $B - A$.

उत्तर एवं व्याख्या देखें

उत्तर: $A - B = \{1, 2\}$ and $B - A = \{5, 6\}$.
A - B = {1, 2}; B - A = {5, 6}.

What is the power set of the empty set $\emptyset$?

उत्तर एवं व्याख्या देखें

उत्तर: $P(\emptyset) = \{\emptyset\}$, which has $2^0 = 1$ element.
P(∅) = {∅} with 1 element.

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