Question

Q#1.. Let P = {1, 3, 5, 7} Q = {3, 7, 9, 11} R = {1, 5, 8, 11}, then verify the following.
(a) P ∪ Q = Q ∪ P
(b) (P ∪ Q) ∪ R = P ∪ (Q ∪ R) (c) P ∩ Q = Q ∩ P
(d) (P ∩ Q) ∩ R = P ∩ (Q ∩ R)
(e) P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R) (f) P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R)

Answer 1

Answer:

Let P = {1, 3, 5, 7} Q = {3, 7, 9, 11} R = {1, 5, 8, 11}

(a) P ∪ Q = Q ∪ P

P ∪ Q = { 1 , 3 , 5 , 7 , 9 ,11}

Q ∪ P = { 1 , 3 , 5 , 7 , 9 ,11}

therefore

P ∪ Q = Q ∪ P

HENCE VERIFIED

(b) (P ∪ Q) ∪ R = P ∪ (Q ∪ R)

(P ∪ Q) ∪ R = { 1 , 3 , 5 , 7 ,8 , 9 ,11}

P ∪ (Q ∪ R) ={ 1 , 3 , 5 , 7 ,8 , 9 ,11}

therefore

(P ∪ Q) ∪ R = P ∪ (Q ∪ R)

HENCE VERIFIED

(c) P ∩ Q = Q ∩ P

P ∩ Q = { 3 , 7 }

Q ∩ P ={ 3 , 7 }

therefore

P ∩ Q = Q ∩ P

HENCE VERIFIED

(d) (P ∩ Q) ∩ R = P ∩ (Q ∩ R)

(P ∩ Q) ∩ R = { }

P ∩ (Q ∩ R) = { }

therefore

(P ∩ Q) ∩ R = P ∩ (Q ∩ R)

HENCE VERIFIED

(e) P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)

P ∪ (Q ∩ R) =  { 1 , 3 , 5 , 7 ,11}

(P ∪ Q)  = { 1 , 3 , 5 , 7 , 9 ,11}

(P ∪ R)  = { 1 , 3 , 5 , 7 , 8 ,11}

(P ∪ Q) ∩ (P ∪ R) = { 1 , 3 , 5 , 7 ,11}

therefore

P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)

HENCE VERIFIED

(f) P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R)

(Q ∪ R) = { 1 , 3 , 5 , 7 , 8 , 9 ,11}

P ∩ (Q ∪ R) = { 1 , 3 , 5 , 7 }

(P ∩ Q)  = { 3 , 7 }

(P ∩ R)  = { 1 ,  5 }

(P ∩ Q) ∪ (P ∩ R) = { 1 , 3 , 5 , 7 }

therefore

P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R)

HENCE VERIFIED

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Answer 2

Answer:

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