﻿ Class 11 NCERT Math Solution
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TOPICS
Formulae

Methods of Sets :-
```1. Roster Method : {1, 2, 3, 4, 5} in braces form.
2. Set-builder Method : {x : x is vowels of english alphabets}.
```
Types of Sets :-
```1. Empty Sets : A = { }  or φ
2. Equal Sets : A = {a, b, c}, B = {c, b, a} then A = B.
3. Finite Sets : A = {4, 3, 5, 6}
4. Infinite Sets : A = {1, 2, 3, 4,...}
5. Subsets : A ⊂ B or B ⊂ A
6. Power Sets : P(A) = 2ⁿ
7. Equivalent Sets : A = {1, 2, 3}, B = {a, b, c} then equivalent.
8. Universal Sets : U = {  }
```
Operations of Sets :-
```1. Union of Sets : A U B
2. Intersection of Sets : A ∩ B
3. Differece of Sets : A - B = { x : x ∈ A and x ∉ B } or B - A =  { x : x ∉ A and x ∈ B }
4. Complement of Sets : A' = U - A
5. Symmetric Difference of Sets :  A ∆ B = (A - B) U (B - A)
6. Disjoint of Sets : A ≠ B
```
Intervals :-
```1. [ a, b ) = {x : a ≤ x < b} - Open Interval
2. ( a, b ] = { x : a < x ≤  b } - Open interval
3. ( a, b ) = {x : a < x < b} - Open Interval
4. [ a, b ] = {x : a ≤ x ≤ b} - Closed Interval
```
Properties of Union :-
```1. Commutative Law : A ∪ B  = B ∪ A
2. Associative Law : (A ∪ B) ∪ C = A ∪ (B ∪ C)
3. Identity Law : A ∪ φ = A
4. Idempotent law : A ∪ A  = A
5. Law of U: U ∪ A  = U
```
Properties of Intersection :-
```1. Commutative Law : A ∩ B  = B ∩ A
2. Associative Law : (A ∩ B) ∩ C = A ∩ (B ∩ C)
3. Identity Law : A ∩ φ = φ
4. Idempotent law : A ∩ A  = A
5. Law of U: U ∩ A  = A
6. Distributive law : A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C )
```
Properties of Complements :-
```1. Complement laws : (a) A ∪ A′  = U,  (b) A ∩ A′ = φ
2. De Morgan’s law : (a) (A ∪ B)' = A′ ∩ B′,  (b) (A ∩ B)′ = A′ ∪ B′
3. Law of double complementation : (A′)′ = A
4. Laws of empty set  :  φ′ = U
5. Law of U: U′ = φ
```
Practical Problems :-
```1. n ( A ∪ B ) = n ( A ) + n ( B )
2. n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )
3. n ( A ∪ B) = n ( A – B) + n ( A  ∩ B ) + n ( B – A )
4. n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A  ∩ B ) – n ( B  ∩ C) – n ( A  ∩ C ) + n ( A  ∩ B  ∩ C )
5. n (A′ ∩ B′) = n (A ∪ B)′ = n (U) – n (A ∪ B) = n (U) – n (A) – n (B) + n (A ∩ B)
6. n ( A - B ) = n ( A ) - n ( A ∩ B )
7. n ( B - A ) = n ( B ) – n ( A ∩ B )
8. n ( A ∩ B ) = n ( B ) - n ( B – A)
9. n ( A ∩ B ) = n ( A ) - n ( A – B)
10. n ( A ) = n ( A – B) + n ( A ∩ B )
11. n ( B ) = n ( B – A) + n ( A ∩ B )
```
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