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
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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)
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