Question

6. Let A = {1 € W x<2}, B = {x eNi<< < 4) and C = {3,5). Verify that
(i) AX(BUC) = (A x B)U(AXC) (ii) Ax(BNC)=(A x B) n(AXC)
(iii) (AUB)X = (AXC) U(BXC)​

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Answer.
Don't give sham answer. ​

Answer 1

Answer:

525453+2114550093++32150093+214093+21133247096=

Answer 2

A = {x ∈ W | x < 2} = {0,1} B = {x ∈ N | 1 < x < 4} = {2,3,4} C = {3,5} (i) LHS = A x (B ∪ C)

B ∪ C = {2, 3, 4} ∪ {3, 5}

= {2, 3, 4, 5} A x (B ∪ C)

= {(0, 2), (0, 3), (0,4), (0, 5),

(1, 2) , (1, 3), (1, 4),(1, 5)} … (1)

RHS = (A x B) ∪ (A x C) (A x B)

= {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)} (A x C)

= {(0, 3), (0, 5), (1, 3), (1, 5)} (A x B) ∪ (A x C)

= {(0, 2), (0, 3), (0,4), (1, 2), (1, 3), (1, 4), (0, 5), (1, 5)} … (2) (1)

= (2), LHS = RHS Hence it is proved.

(ii) A x (B ∩ C) = (A x B) ∩ (A x C)

LHS = A x (B ∩ C) (B ∩ C)

= {3} A x (B ∩ C) = {(0, 3), (1, 3)} … (1)

RHS = (A x B) ∩ (A x C) (A x B)

= {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)} (A x C) = {(0, 3), (0, 5), (1, 3), (1, 5)} (A x B) ∩ (A x C)

= {(0, 3), (1, 3)} ... (2) (1) = (2)

⇒ LHS = RHS. Hence it is verified.

(iii) (A ∪ B) x C = (A x C) ∪ (B x C)

LHS = (A ∪ B) x C A ∪ B

= {0, 1, 2, 3, 4} (A ∪ B) x C

= {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} … (1)

RHS = (A x C) ∪ (B x C) (A x C)

= {(0, 3), (0, 5), (1, 3), (1, 5)} (B x C)

= {(2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} (A x C) ∪ (B x C)

= {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} … (2) (1) = (2) ∴ LHS = RHS. Hence it is verified.