VERIFYING COMMUTATIVE AND ASSOCIATIVE PROPERTIES WITH GIVEN SETS

Question 1 :

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

AUB  =  BUA

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

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

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

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

(1)  =  (2)

Question 2 :

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)

Question 3 :

For A  =  {x|x is a prime factor of 42}, B  =  {x|5 < x ≤ 12, x ∈ N} and C  =  {1, 4, 5, 6}

verify AU(BUC)  =  (AUB)UC.

Solution :

A  =  {x|x is a prime factor of 42}

A  =  {2, 3, 7}

B  =  {x|5 < x ≤ 12, x ∈ N}

B  =  {6, 7, 8, 9, 10, 11, 12}

C  =  {1, 4, 5, 6}

L.H.S

AU(BUC)

(BUC)  =  {6, 7, 8, 9, 10, 11, 12}U{1, 4, 5, 6}

  =  {1, 4, 5, 6, 7, 8, 9, 10, 11, 12}

AU(BUC)  =  {2, 3, 7}U{1, 4, 5, 6, 7, 8, 9, 10, 11, 12}

  =  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}  --- (1)

R.H.S

(AUB)UC

(AUB)  =  {2, 3, 7}U{6, 7, 8, 9, 10, 11, 12}

(AUB)  = {2, 3, 6, 7, 8, 9, 10, 11, 12}

(AUB)UC  =  {2, 3, 6, 7, 8, 9, 10, 11, 12} U {1, 4, 5, 6}

(AUB)UC  =  {1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12}  ----(2)

(1)  =  (2) 

Question 4 :

Given P = {a, b, c, d, e} Q = {a, e, i, o, u} and R ={a, c, e, g}. Verify the associative property of set intersection.

Solution :

P∩(Q∩R)  =  (P∩Q)∩R

L.H.S

P∩(Q∩R)

(Q∩R)  =  {a, e, i, o, u} ∩ {a, c, e, g}

Q∩R  =  {a, e} 

P∩(Q∩R)  =  {a, b, c, d, e}∩{a, e}

       P∩(Q∩R)  =  {a, e} ---- (1)

R.H.S

(P∩Q)∩R

(P∩Q)  =  {a, b, c, d, e} ∩ {a, e, i, o, u}

(P∩Q)  =  {a, e}

(P∩Q)∩R  =  {a, e}∩{a, c, e, g}   

(P∩Q)∩R  =  {a, e} ---- (2)

(1)  =  (2)

Hence proved 

Question 5 :

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

(i) (A ∪ B) ∩ (A ∪ C)

(ii) (A ∩ B) ∪ (A ∩ C)

(iii) (A ∪ B ∪ C)′

(iv) (A ∪B′) ∩ (A′ ∪ B)

Solution :

(i)

(A ∪ B) = {1, 2, 3, 4, 5} U {4, 5, 6, 7}

= {1, 2, 3, 4, 5, 6, 7}

(A ∪ C) = {1, 2, 3, 4, 5} ∪ {5, 6, 7, 8}

= {1, 2, 3, 4, 5, 6, 7, 8}

Therefore,

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

(ii) (A ∩ B) = {1, 2, 3, 4, 5} ∩ {4, 5, 6, 7}

= {4, 5}

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

= {5}

Therefore,

(A ∩ B) ∪ (A ∩ C) = {4, 5} ∪ {5}

= {4, 5}

(iii) (A ∪ B ∪ C)′ = U – (A ∪ B ∪ C)

But,

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

Therefore,

(A ∪ B ∪ C)′

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 5, 6, 7, 8}

= {9}

(iv) (A ∪ B′) = {1, 2, 3, 4, 5} ∪ {1, 2, 3, 8, 9}

= {1, 2, 3, 4, 5, 8, 9}

And (A′ ∪ B) = {6, 7, 8, 9} ∪ {4, 5, 6, 7, 8}

= {4, 5, 6, 7, 8, 9}

Therefore, (A ∪ B′) ∩ (A′ ∪ B)

= {1, 2, 3, 4, 5, 8, 9} ∩ {4, 5, 6, 7, 8, 9}

= {4, 5, 8, 9}

Question 6 :

If Set A = {1, 2, 3, 5, 6} and Set B = {1, 3, 4, 8, 9} then verify that A ∆ B = B ∆ A and also prove that A ∆ B = (A ∪ B) – (A ∩ B) ? 

Solution :

A ∆ B = (A – B) ∪ (B – A)

    = {2, 5, 6} ∪ {4, 8, 9} = {2, 4, 5, 6, 8, 9}

B ∆ A = (B – A) ∪ A – B

= {4, 8, 9} ∪ {2, 5, 6}

= {2, 4, 5, 6, 8, 9}

Similarly, in order to prove that

A ∆ B = (A ∪ B) – (A ∩ B)

As (A ∪ B) = {1, 2, 3, 5, 6} ∪ {1, 3, 4, 8, 9}

= {1, 2, 3, 4, 5, 6, 8, 9}

And (A ∩ B) = {1, 3}

(A ∪ B) – (A ∩ B) = {1, 2, 3, 4, 5, 6, 8, 9} – {1, 3}

= {2, 4, 5, 6, 8, 9}

Hence A ∆ B = (A ∪ B) – (A ∩ B)

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