29) Let A = {x ∈ W | x < 2}, B = {x ∈ N | 1 < x ≤ 4} and C = {3,5}. Verify that A x (B U C) = (A x B) U (A x C)
Answer: Given A = {x ∈ W | x < 2} ⇒ A = {0,1}
B = {x ∈ N | 1 < x ≤ 4} ⇒ B = {2,3,4}
C = {3,5}
To verify: A x (B U C) = (A x B) U (A x C)
LHS:
B U C = {2,3,4} U {3,5} = {2,3,4,5}
A x (B U C) = {0,1} x {2,3,4,5} = {(0,2), (0,3), (0,4), (0,5), (1,2), (1,3), (1,4), (1,5)} ... (1)
RHS:
A x B = {0,1} x {2,3,4} = {(0,2), (0,3), (0,4), (1,2), (1,3), (1,4)}
A x C = {0,1} x {3,5} = {(0,3), (0,5), (1,3), (1,5)}
(A x B) U (A x C) = {(0,2), (0,3), (0,4), (0,5), (1,2), (1,3), (1,4), (1,5)} ... (2)
From (1) and (2), it is clear that LHS = RHS.
Hence verified.