Answer : Given: A, B and C three sets are given. Need to prove: A × (B ∩ C) = (A × B) ∩ (A × C)
Let us consider, (x, y) A × (B ∩ C)
| ⇒ x | A and y | (B ∩ C) | |
| ⇒ x
⇒ (x |
A and (y
A and y |
B and y C)
B) and (x A and y |
C) |
⇒ (x, y) (A × B) and (x, y) (A × C)
⇒ (x, y) (A × B) ∩ (A × C)
From this we can conclude that,
⇒ A × (B ∩ C) ⊆ (A × B) ∩ (A × C)—- (1)
Let us consider again, (a, b) (A × B) ∩ (A × C)
⇒ (a, b) (A × B) and (a, b) (A × C)
⇒ (a A and b B) and (a A and b C)
⇒ a A and (b B and b C)
⇒ a A and b (B ∩ C)
⇒ (a, b) A × (B ∩ C)
From this, we can conclude that,
⇒ (A × B) ∩ (A × C) ⊆ A × (B ∩ C)—- (2)
Now by the definition of the set we can say that, from (1) and (2), A × (B ∩ C) = (A × B) ∩ (A × C) [Proved]