Let A = {1,2,3}, B = {2,3,4}, C = {4,5} and D = {5,6,7,8}Verify that :(i) A × (B∩C) = (A×B)∩(A×C)(ii) A × C is a subset of B × D
Question
Let A = {1,2,3}, B = {2,3,4}, C = {4,5} and D = {5,6,7,8}Verify that :(i) A × (B∩C) = (A×B)∩(A×C)(ii) A × C is a subset of B × D
Solution
(i) A × (B∩C) = (A×B)∩(A×C)
First, we need to find the intersection of sets B and C, which is B∩C = {4}.
Then, we find the Cartesian product of A and (B∩C), which is A × (B∩C) = {(1,4), (2,4), (3,4)}.
Next, we find the Cartesian products of A and B, and A and C, which are A×B = {(1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,2), (3,3), (3,4)} and A×C = {(1,4), (1,5), (2,4), (2,5), (3,4), (3,5)} respectively.
The intersection of these two sets, (A×B)∩(A×C), is {(1,4), (2,4), (3,4)}.
Therefore, A × (B∩C) = (A×B)∩(A×C).
(ii) A × C is a subset of B × D
The Cartesian product of A and C is A×C = {(1,4), (1,5), (2,4), (2,5), (3,4), (3,5)}.
The Cartesian product of B and D is B×D = {(2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8)}.
We can see that every pair in A×C is also in B×D. Therefore, A × C is a subset of B × D.
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