Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that: $\text{A}\times(\text{B}\cap\text{C})=(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$
✓
Answer
We have, A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8} $\therefore\text{ B}\cap\text{C}=\{1,2,3,4\}\cap\{5,6\}=\phi$ $\text{A}\times(\text{B}\cap\text{C})=\{1,2\}\times\phi=\phi\ ...(\text{i})$ Now, A × B = {1, 2} × {1, 2, 3, 4} = {(1,1), (1, 2), (1,3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)} and, A × C = {1, 2} × {5, 6} = {(1, 5), (1, 6), (2, 5), (2, 6)} $\therefore\ (\text{A}\times\text{B})\cap(\text{A}\times\text{C})=\phi\ ...(\text{ii})$ From equation (i) and equation (ii), we get $\text{A}\times(\text{B}\cap\text{C})=(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$ Hence verified.
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