Let A={1,2,3}, B={4} and C ={5}
Verify that A×(B-C)=(A×B)-(A×C)?
find (A×B) intersection (A×C)
Answers
Given as
A = {1, 2, 3}, B = {4} and C = {5}
(i) A × (B ∪ C) = (A × B) ∪ (A × C)
Let us consider the LHS (B ∪ C)
(B ∪ C) = {4, 5}
A × (B ∪ C) = {1, 2, 3} × {4, 5}
= {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
Then, RHS
(A × B) = {1, 2, 3} × {4}
= {(1, 4), (2, 4), (3, 4)}
(A × C) = {1, 2, 3} × {5}
= {(1, 5), (2, 5), (3, 5)}
(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)
Let us consider the LHS: (B ∩ C)
(B ∩ C) = ∅ (Here, no common element)
A × (B ∩ C) = {1, 2, 3} × ∅
= ∅
Then, RHS
(A × B) = {1, 2, 3} × {4}
= {(1, 4), (2, 4), (3, 4)}
(A × C) = {1, 2, 3} × {5}
= {(1, 5), (2, 5), (3, 5)}
(A × B) ∩ (A × C) = ∅
∴ LHS = RHS
(iii) A × (B − C) = (A × B) − (A × C)
Let us consider the LHS: (B − C)
(B − C) = ∅
A × (B − C) = {1, 2, 3} × ∅
= ∅
Then, RHS
(A × B) = {1, 2, 3} × {4}
= {(1, 4), (2, 4), (3, 4)}
(A × C) = {1, 2, 3} × {5}
= {(1, 5), (2, 5), (3, 5)}
(A × B) − (A × C) = ∅
Thus, LHS = RHS