If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm = (mn/2)(1 + mn)

If S1, S2, S3,....Sm are the sums of n terms of m A.P.’s whose first terms are 1,2, 3,...m and whose common differences are 1, 3, 5,..., (2m -1) respectively, then show that S1 + S2 + S3 +............Sm  = (mn/2)(1 + mn)

Solution :

n = [(l-a)/d] + 1

 n  = [((2mn - n + 1) - (1 + n))/2n] + 1

 n  = [(2mn - n + 1 - 1 - n)/2n] + 1

n = [(2mn - 2n)/2n] + 1

n = (m - 1) + 1

n = m

 S1 + S2 + S3 +............Sm  

Hence proved.

Question 3: Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. Chapter 1 - Relations And Functions

Chapter 1 - Relations And FunctionsNCERT Solutions for Class 12 Science Math

Question 3:

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} asR = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

ANSWER:

Let A = {1, 2, 3, 4, 5, 6}.

A relation R is defined on set A as:

R = {(a, b): b = a + 1}

∴R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}

We can find (a, a) ∉ R, where a ∈ A.

For instance, 

(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R

∴R is not reflexive.

It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R.

∴R is not symmetric.

Now, (1, 2), (2, 3) ∈ R

But, 

(1, 3) ∉ R

∴R is not transitive 

Hence, R is neither reflexive, nor symmetric, nor transitive.

For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that (i) A U (B ∩ C) = (A U B) ∩ (A U C) (ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

(11) For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

(iii) Verify using Venn diagrams

Solution:

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

L.H.S

A U (B ∩ C)

(B ∩ C) = {-1,-2,3,4,5,6} ∩ {-1,2,3,4,5,7}

       = {-1,3,4,5}

A U (B ∩ C) = {-3,-1,0,4,6,8,10} U {-1,3,4,5}

            = {-3,-1,0,3,4,5,6,8,10}  --- (1)

R.H.S

(A U B) ∩ (A U C)

(A U B) = {-3,-1,0,4,6,8,10} U {-1,-2,3,4,5,6}

        = {-3,-2,-1,0,3,4,5,6,8,10}

(A U C) = {-3,-1,0,4,6,8,10} U {-1,2,3,4,5,7}

       = {-3,-1,0,2,3,4,5,6,7,8,10}

(AUB)∩(AUC)={-3,-2,-1,0,3,4,5,6,8,10} ∩ {-3,-1,0,2,3,4,5,6,7,8,10}

             = {-3,-1,0,3,4,5,6,8,10}  --- (2)

(1) = (2)

A U (B ∩ C) = (A U B) ∩ (A U C)

Hence proved

---

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7}

L.H.S

(B U C) = {-1,-2,3,4,5,6} U {-1,2,3,4,5,7}

       = {-2,-1,2,3,4,5,6,7}

A ∩ (B U C) = {-3,-1,0,4,6,8,10} ∩ {-2,-1,2,3,4,5,6,7}

            = {-1,4,6} ---- (1)

R.H.S

(A ∩ B) = {-3,-1,0,4,6,8,10} ∩ {-1,-2,3,4,5,6}

       = {-1,4,6}

(A ∩ C) = {-3,-1,0,4,6,8,10} ∩ {-1,2,3,4,5,7}

       = {-1,4}

(A ∩ B) U (A ∩ C) = {-1,4,6} U {-1,4}

                    = {-1,4,6} ---- (2)

(1) = (2)

A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Hence proved

Verify the commutative property of set intersection for A = {l,m,n,o,2,3,4,7} and B = {2,5,3,-2,m,n,o,p}

Question 6

Verify the commutative property of set intersection for A = {l,m,n,o,2,3,4,7} and B = {2,5,3,-2,m,n,o,p}

Solution:

commutative property of set intersection

A ⋂ B = B ⋂ A

A ⋂ B = {l,m,n,o,2,3,4,7} ⋂ {2,5,3,-2,m,n,o,p}

     = {m,n,o}  --- (1)

B ⋂ A = {2,5,3,-2,m,n,o,p} ⋂ {l,m,n,o,2,3,4,7}

     = {m,n,o}  --- (2)

(1) = (2)

Given A = {a,x,y,r,s}, B = {1,3,5,7,-10},verify the commutative property of set union.

Question 5

Given A = {a,x,y,r,s}, B = {1,3,5,7,-10},verify the commutative property of set union.

Solution:

commutative property of set union

A U B = B U A

A U B = {a,x,y,r,s} U {1,3,5,7,-10}

    =  {a,x,y,r,s,1,3,5,7,-10}   ------ (1)

B U A = {1,3,5,7,-10} U {a,x,y,r,s}

    =  {a,x,y,r,s,1,3,5,7,-10}  ------ (2)

(1) = (2)

If A = {4, 6, 7, 8, 9} , B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6}, then find (i) A U (B ∩ C) (ii) A ∩ (B U C) (iii) A \ (C \ B)

Question 4 :

If A = {4, 6, 7, 8, 9} , B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6}, then find

(i) A U (B ∩ C) (ii) A ∩ (B U C)    (iii) A \ (C \ B)

Solution :

(i) A U (B ∩ C)

(B ∩ C) = {2, 4, 6} ∩ {1, 2, 3, 4, 5, 6}

 = {2, 4, 6}

A U (B ∩ C) = {4, 6, 7, 8, 9} U {2, 4, 6}

 = {2, 4, 6, 7, 8, 9}

(ii) A ∩ (B U C)

(B U C) = {2 ,4, 6} U {1, 2, 3, 4, 5, 6}

       = {1, 2, 3, 4, 5, 6}

A ∩ (B U C) = {4, 6, 7, 8, 9} ∩ {1, 2, 3, 4, 5, 6}

            = {4, 6}

(iii) A \ (C \ B)

C \ B = {1, 2, 3, 4, 5, 6} \ {2, 4, 6}

    = {1, 3, 5}

A \ (C \ B) = {4, 6, 7, 8, 9} \ {1, 3, 5}

           = {4, 6, 7, 8, 9}