Class 11 || Sets || Laws of Algebra of Sets

1. Commutative Laws

For any two finite sets A and B;

1. A U B = B U A
2. A ∩ B = B ∩ A

2. Associative Laws

For any three finite sets A, B and C;

1. (A U B) U C = A U (B U C)
2. (A ∩ B) ∩ C = A ∩ (B ∩ C)

Thus, union and intersection are associative.

3. Idempotent Laws

For any finite set A;

1. A U A = A
2. A ∩ A = A

4. Distributive Laws

For any three finite sets A, B and C;

1. A U (B ∩ C) = (A U B) ∩ (A U C)
2. A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Thus, union and intersection are distributive over intersection and union respectively.

5. De Morgan’s Laws

 For any two finite sets A and B;

1. A – (B U C) = (A – B) ∩ (A – C)
2. A - (B ∩ C) = (A – B) U (A – C)

De Morgan’s Laws can also we written as:

1. (A U B)’ = A' ∩ B'
2. (A ∩ B)’ = A' U B'

More laws of algebra of sets:

6. For any two finite sets A and B

1. A – B = A ∩ B'
2. B – A = B ∩ A'
3. A – B = A ⇔ A ∩ B = ∅
4. (A – B) U B = A U B
5. (A – B) ∩ B = ∅
6. A ⊆ B ⇔ B' ⊆ A'
7. (A – B) U (B – A) = (A U B) – (A ∩ B)

7. For any three finite sets A, B and C

1. A – (B ∩ C) = (A – B) U (A – C)
2. A – (B U C) = (A – B) ∩ (A – C)
3. A ∩ (B - C) = (A ∩ B) - (A ∩ C)
4. A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)

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Class 10 || Chapter 01 : Real Numbers || Question Section - A

Questions :

1. Show that every positive even integer is of the form 2q and that every positive odd integer is of the form 2q+1 , where q is some integer. ( NCERT )
   
2. Show that any positive integer is of the form 3q or, 3q + 1 or, 3q + 2 some integer q.
   
3. Show that any positive odd integer is of the form 4q + 1 or 4q + 3 where q is some integer. ( NCERT )
   
4. Show that the square of an odd integer is of the form 4q + 1 for the some integer q. ( NCERT EXEMPLAR )
   
5. Show that n square - 1 is divisible by, 8 if n is an odd positive integer. ( NCERT EXEMPLER )
   
6. Prove that if x and y are are odd positive integer, x squared plus y squared is even but not divisible by 4. HINT : let x = 2m + 1 and y = 2n + 1 where m & n is integer. (R.D. SHARMA)
   
7. Prove that n square - n is divisible by 2 for every positive integer n. HINT : integer is of the form 2q or 2q + 1.
   
8. Show that the square of any positive integer is of the form 3m or 3m + 1 for some integer m. ( NCERT )
   
9. use euclid's division lemma to show that the cube of any positive integer is is either of the form 9m, 9m+ 1 or 9 m + 8 for some integer m. ( NCERT )
   
10. show that the cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3 for some integer m.
    
11. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
    
12. Show that one and only one out of n, n + 2 and n + 4 is divisible by 3 where n is positive integer. (R.D. SHARMA)
    
13. Prove that one of every three consecutive positive integer is divisible by 3.
    
14. prove that the product of two consecutive positive integers is divisible by 2.
    
15. Prove that the product of three consecutive positive integers is divisible by 6.
    
16. For any positive integer n, prove that n cube - n is divisible by 3.
    
17. Prove that if a positive integer is of the form 6q + 5 then, it is of the form 3q + 2 for some integer q but not conversely.