QUICK REVISION NOTES & FORMULA SHEET

Class 11 Mathematics

Chapter: Sets

### 1. Introduction to Sets

• Set: A well-defined collection of objects. (समुच्चय: वस्तुओं का एक सुपरिभाषित संग्रह।)
• Representation of Sets:
  • Roster Form (सारणीबद्ध रूप): Elements are listed inside braces, separated by commas. Example: $A = {1, 2, 3, 4}$
  • Set-Builder Form (समुच्चय निर्माण रूप): Elements are described by a characterizing property. Example: $A = {x : x \in \mathbb{N}, 1 \le x \le 4}$

### 2. Types of Sets (समुच्चयों के प्रकार)

• Empty Set / Null Set / Void Set ($\phi$ or {}): A set which contains no elements. (रिक्त समुच्चय)
• Finite Set: A set with a definite/countable number of elements. (परिमित समुच्चय)
• Infinite Set: A set with an infinite number of elements. (अपरिमित समुच्चय)
• Equal Sets: Two sets $A$ and $B$ having exactly the same elements ($A = B$). (समान समुच्चय)
• Equivalent Sets: Two sets having the same number of cardinal elements ($n(A) = n(B)$). (समरूप समुच्चय)

### 3. Subsets and Intervals (उपसमुच्चय और अंतराल)

• Subset ($A \subseteq B$): Every element of $A$ is also an element of $B$. ($A$ समुच्चय $B$ का उपसमुच्चय है।)
• Proper Subset ($A \subset B$): $A \subseteq B$ and $A \neq B$.
• Power Set $P(A)$: The collection of all subsets of a set $A$. If $n(A) = m$, then $n(P(A)) = 2^m$. (घात समुच्चय)
• Universal Set ($U$): A set containing all objects under consideration. (सार्वत्रिक समुच्चय)
• Intervals as Subsets of $\mathbb{R}$ (वास्तविक संख्याओं کے उपसमुच्चय के रूप में अंतराल):
  • Open Interval: $(a, b) = {x : a < x < b}$
  • Closed Interval: $[a, b] = {x : a \le x \le b}$
  • Semi-open / Semi-closed: $[a, b) = {x : a \le x < b}$ or $(a, b] = {x : a < x \le b}$

### 4. Venn Diagrams & Operations on Sets (वेन आरेख और समुच्चयों पर संक्रियाएँ)

• Union of Sets ($A \cup B$): ${x : x \in A \text{ or } x \in B}$ (सम्मेलन)
• Intersection of Sets ($A \cap B$): ${x : x \in A \text{ and } x \in B}$ (सर्वनिष्ठ)
• Disjoint Sets: If $A \cap B = \phi$, then $A$ and $B$ are disjoint sets. (असंयुक्त समुच्चय)
• Difference of Sets ($A - B$ or $A \setminus B$): ${x : x \in A \text{ and } x \notin B}$ (अंतर समुच्चय)
• Complement of a Set ($A'$ or $A^c$): $U - A$ or ${x : x \in U \text{ and } x \notin A}$ (पूरक समुच्चय)

### 5. Laws of Algebra of Sets (समुच्चय बीजगणित के नियम)

1. Idempotent Laws (वर्गसम नियम):
   • $A \cup A = A$
   • $A \cap A = A$
2. Identity Laws (तत्समक नियम):
   • $A \cup \phi = A$
   • $A \cap U = A$
3. Commutative Laws (क्रमविनिमय नियम):
   • $A \cup B = B \cup A$
   • $A \cap B = B \cap A$
4. Associative Laws (साहचर्य नियम):
   • $(A \cup B) \cup C = A \cup (B \cup C)$
   • $(A \cap B) \cap C = A \cap (B \cap C)$
5. Distributive Laws (बंटन नियम):
   • $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
   • $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
6. De Morgan's Laws (डी मॉर्गन नियम - Very Important):
   • $(A \cup B)' = A' \cap B'$
   • $(A \cap B)' = A' \cup B'$

### 6. Important Cardinality Formulas (महत्वपूर्ण संख्यात्मक सूत्र)

For finite sets $A$, $B$, and $C$:

• $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
• If $A$ and $B$ are disjoint ($A \cap B = \phi$): $n(A \cup B) = n(A) + n(B)$
• $n(A - B) = n(A) - n(A \cap B)$
• $n(A' ) = n(U) - n(A)$
• $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$