Question -
Answer -
Given:
A = {1, 2, 3}, B = {4} and C = {5}
(i) A ├Ч (B тИк C) = (A ├Ч B) тИк (A ├Ч C)
Let us consider LHS: (B тИк C)
(B тИк C) = {4, 5}
A ├Ч (B тИк C) = {1, 2, 3} ├Ч {4, 5}
= {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
Now, RHS
(A ├Ч B) = {1, 2, 3} ├Ч {4}
= {(1, 4), (2, 4), (3, 4)}
(A ├Ч C) = {1, 2, 3} ├Ч {5}
= {(1, 5), (2, 5), (3, 5)}
(A ├Ч B) тИк (A ├Ч C) = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)}
тИ┤ LHS = RHS
(ii) A ├Ч (B тИй C) = (A ├Ч B) тИй (A ├Ч C)
Let us consider LHS: (B тИй C)
(B тИй C) = тИЕ (No common element)
A ├Ч (B тИй C) = {1, 2, 3} ├Ч тИЕ
= тИЕ
Now, RHS
(A ├Ч B) = {1, 2, 3} ├Ч {4}
= {(1, 4), (2, 4), (3, 4)}
(A ├Ч C) = {1, 2, 3} ├Ч {5}
= {(1, 5), (2, 5), (3, 5)}
(A ├Ч B) тИй (A ├Ч C) = тИЕ
тИ┤ LHS = RHS
(iii) A ├Ч (B тИТ C) = (A ├Ч B) тИТ (A ├Ч C)
Let us consider LHS: (B тИТ C)
(B тИТ C) = тИЕ
A ├Ч (B тИТ C) = {1, 2, 3} ├Ч тИЕ
= тИЕ
Now, RHS
(A ├Ч B) = {1, 2, 3} ├Ч {4}
= {(1, 4), (2, 4), (3, 4)}
(A ├Ч C) = {1, 2, 3} ├Ч {5}
= {(1, 5), (2, 5), (3, 5)}
(A ├Ч B) тИТ (A ├Ч C) = тИЕ
тИ┤ LHS = RHS