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4. If \( A=\{5,6\}, B=\{4,5,6\}, C=\{5,6,7\} \), Show that \( A \times A=(B \times B) \cap(C \times C) \).

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We have A = {5, 6}, B = {4, 5, 6} and C = {5, 6, 7}

Now, A X A = {5, 6} X {5, 6}

= {(5, 5), (5, 6), (6, 5), (6, 6)}

And B X B = {(4, 4), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}

Also C X C = {(5, 5), (5, 6), (5, 7), (6, 5), (6, 6), (6, 7), (7, 5), (7, 6), (7, 7)}

Therefore, (B X B) ∩ (C X C) = {(5, 5), (5, 6), (6, 5), (6, 6)}

= A X A

Hence Proved

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