If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:
(i) A × (B ∪ C) = (A × B) ∪ (A × C)
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
(iii) A × (B − C) = (A × B) − (A × C)
Given:
A = {1, 2, 3}, B = {4} and C = {5}
(i) A × (B ∪ C) = (A × B) ∪ (A × C)
We have:
(B ∪ C) = {4, 5}
LHS: A × (B ∪ C) = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
Now,
(A × B) = {(1, 4), (2, 4), (3, 4)}
And,
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: (A × B) ∪ (A × C) = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)}
∴ LHS = RHS
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
We have:
$(B \cap C)=\phi$
$\mathrm{LHS}: A \times(B \cap C)=\phi$
And,
(A × B) = {(1, 4), (2, 4), (3, 4)}
(A × C) = {(1, 5), (2, 5), (3, 5)}
$\mathrm{RHS}:(A \times B) \cap(A \times C)=\phi$
∴ LHS = RHS
(iii) A × (B − C) = (A × B) − (A × C)
We have:
$(B-C)=\phi$
$\mathrm{LHS}: A \times(B-C)=\phi$
Now,
(A × B) = {(1, 4), (2, 4), (3, 4)}
And,
(A × C) = {(1, 5), (2, 5), (3, 5)}
RHS: $(A \times B)-(A \times C)=\phi$
∴ LHS = RHS
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.