Given A = {x ∈ N : x 6}, B = {3, 6, 9} and C = {x ∈ N : 2x \- 5 ≤ 8}. Show that : (i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Question

KnowledgeBoat · Accepted Answer

A = {x ∈ N : x 6} A = {1, 2, 3, 4, 5} B = {3, 6, 9} C = {x ∈ N : 2x \- 5 ≤ 8} 2x \- 5 ≤ 8 2x ≤ 8 \+ 5 2x ≤ 13 x ≤ x ≤ 6.5 C = {1, 2, 3, 4, 5, 6} (i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) Taking LHS: A ∪ (B ∩ C) B ∩ C - contains all the common elements in set B and C. B ∩ C = {3, 6, 9} ∩ {1, 2, 3, 4, 5, 6} B ∩ C = {3, 6} A ∪ (B ∩ C) - contains all the elements in set A and B ∩ C. A ∪ (B ∩ C) = {1, 2, 3, 4, 5} ∪ {3, 6} A ∪ (B ∩ C) = {1, 2, 3, 4, 5, 6} .............(1) Taking RHS : (A ∪ B) ∩ (A ∪ C) A ∪ B - contains all the elements in set A and B. A ∪ B = {1, 2, 3, 4, 5} ∪ {3, 6, 9} A ∪ B = {1, 2, 3, 4, 5, 6, 9} A ∪ C - contains all the elements in set A and C. A ∪ C = {1, 2, 3, 4, 5} ∪ {1, 2, 3, 4, 5, 6} A ∪ C = {1, 2, 3, 4, 5, 6} (A ∪ B) ∩ (A ∪ C) - contains all the common elements in set (A ∪ B) and (A ∪ C) (A ∪ B) ∩ (A ∪ C) = {1, 2, 3, 4, 5, 6, 9} ∩ {1, 2, 3, 4, 5, 6} (A ∪ B) ∩ (A ∪ C) = {1, 2, 3, 4, 5, 6} .............(2) From (1) and (2), we get LHS = RHS Hence, A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Taking LHS: A ∩ (B ∪ C) B ∪ C - contains all the elements in set B and C. B ∪ C = {3, 6, 9} ∪ {1, 2, 3, 4, 5, 6} B ∪ C = {1, 2, 3, 4, 5, 6, 9} A ∩ (B ∪ C) - contains all the common elements in set A and B ∪ C. A ∩ (B ∪ C) = {1, 2, 3, 4, 5} ∪ {1, 2, 3, 4, 5, 6, 9} A ∩ (B ∪ C) = {1, 2, 3, 4, 5} .............(3) Taking RHS : (A ∩ B) ∪ (A ∩ C) A ∩ B - contains all the common elements in set A and B. A ∩ B = {1, 2, 3, 4, 5} ∩ {3, 6, 9} A ∩ B = {3} A ∩ C - contains all the common elements in set A and C. A ∩ C = {1, 2, 3, 4, 5} ∩ {1, 2, 3, 4, 5, 6} A ∩ C = {1, 2, 3, 4, 5} (A ∩ B) ∪ (A ∩ C) - contains all the elements in set (A ∩ B) and (A ∩ C) (A ∩ B) ∪ (A ∩ C) = {3} ∪ {1, 2, 3, 4, 5} (A ∩ B) ∪ (A ∩ C) = {1, 2, 3, 4, 5} .............(4) From (3) and (4), we get LHS = RHS Hence, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Mathematics

Given A = {x ∈ N : x < 6}, B = {3, 6, 9} and C = {x ∈ N : 2x - 5 ≤ 8}. Show that :
(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Sets

Related Questions

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