Let A = {a, b, d, e}, B = {b, c, e, f} and C = {d, e, f, g}
1) Verify A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
2) (Verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
1) Verify A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
2) (Verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
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2) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
L.H.S. = A ∩ (B ∪ C)
B ∪ C = {b, c, d, e, f, g}
A ∩ (B ∪ C) = {b, d, e} ... (1)
R.H.S. = (A ∩ B) ∪ (A ∩ C)
A ∩ B = {b, e}
A ∩ C = {d, e}
(A ∩ B) ∪ (A ∩ C) = {b, d, e} … (2)
From (1) and (2), we conclude that;
A ∩ (B ⋃ C) = (A ∩ B) ⋃ (A ∩ C) [verified]
2) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
L.H.S. = A ∪ (B ∩ C)
B ∩ C = {e, f}
A ∪ (B ∩ C) = {a, b, d, e, f} ... (1)
R.H.S. = (A ∪ B) ∩ (A ∪ C)
A∪B = {a, b, c, d, e, f}
A∪C = {a, b, d, e, f, g}
(A ∪ B) ∩ (A ∪ C) = {a, b, d, e, f} ... (2)
From (1) and (2), we conclude that
A ∪ (B ∩ C) = A ∪ B ∩ (A ∪ C) [verified]
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