Commutative Law :
A u B = B u A
A n B = B n A
Associative Law :
A u (B u C) = (A u B) u C
A n (B n C) = (A n B) n C
Distributive Law :
A n (B u C) = (A n B) u (A n C)
A u (B n C) = (A u B) n (A u C)
Identity Law :
AU∅ = A
A n U = A
Complement Law :
AUA' = U
A n A' = ∅
Double Complement Law :
(A')' = A
Idempotent Law :
AUA = A
A n A = A
Universal Bound Law :
AuU = U
An∅ = ∅
De morgan's Law
(AuB)' = A'nB'
(AnB)' = A'uB'
Absorption Law :
Au(AnB) = A
An(AUB) = A
Example 1 :
For the given sets
A = {-10, 0, 1, 9, 2, 4, 5}
B = {-1, -2, 5, 6, 2, 3, 4},
Verify the following :
(i) AuB = BuA
(ii) AnB = BnA
Solution :
(i) Let us verify that union is commutative.
A u B = {-10, 0, 1, 9, 2, 4, 5} u {-1, -2, 5, 6, 2, 3, 4}
A u B = {-10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9} ---------(1)
B u A = {-1, -2, 5, 6, 2, 3, 4 } u {-10, 0, 1, 9, 2, 4, 5}
B u A = {-10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9} ---------(2)
From (1) and (2), we have
A u B = B u A
(ii)
A n B = {-10, 0, 1, 9, 2, 4, 5} n {-1, -2, 5, 6, 2, 3, 4}
A n B = {2, 4, 5} ---------(1)
B n A = {-1, -2, 5, 6, 2, 3, 4 } u {-10, 0, 1, 9, 2, 4, 5}
B n A = {2, 4, 5} ---------(2)
From (1) and (2), we have
A n B = B n A
Example 2 :
For the given sets
A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}
verify that
A u (B u C ) = (A u B) u C
Also verify it by using Venn diagram.
Solution :
Let us verify that set union is associative.
B u C = {3, 4, 5, 6} u {5, 6, 7, 8}
B u C = {3, 4, 5, 6, 7, 8}
A u (B u C) = {1, 2, 3, 4, 5} u {3, 4, 5, 6, 7, 8}
A u (B u C) = {1, 2, 3, 4, 5, 6, 7, 8} ---------(1)
A u B = {1, 2, 3, 4, 5} u {3, 4, 5, 6}
A u B = {1, 2, 3, 4, 5, 6}
(A u B) u C = {1, 2, 3, 4, 5, 6} u {5, 6, 7, 8}
(A u B) u C = {1, 2, 3, 4, 5, 6, 7, 8} ---------(2)
From (1) and (2), we have
A u (B u C) = (A u B) u C
Example 3 :
For the given sets
A = {a, b, c, d}, B = {a, c, e}, C = {a, e}
verify that
A n (B n C) = (A n B) n C
Also verify it by using Venn diagram.
Solution :
Let us verify that set intersection is associative.
B n C = {a, c, e} u {a, e}
B n C = {a, e}
A n (B n C) = {a, b, c, d} n {a, e}
A n (B n C) = {a} ---------(1)
A n B = {a, b, c, d} u {a, c, e}
A n B = {a, c}
(A n B) n C = {a, c} n {a, e}
(A n B) n C = {a} ---------(2)
From (1) and (2), we have
A n (B n C) = (A n B) n C
Example 4 :
For the given sets
A = {0, 1, 2, 3, 4}, B = {1, -2, 3, 4, 5, 6}, C = {2, 4, 6, 7}
verify that
A u (B n C ) = (A u B) n (A u C)
Also verify it by using Venn diagram.
Solution :
Let us verify that union distributes over intersection.
B n C = {1, -2, 3, 4, 5, 6} n {2, 4, 6, 7}
B n C = {4, 6}
A u (B n C) = {0, 1, 2, 3, 4} u {4, 6}
A u (B n C) = {0, 1, 2, 3, 4, 6} -----(1)
A u B = {0, 1, 2, 3, 4} u {1, -2, 3, 4, 5, 6}
A u B = {-2, 0, 1, 2, 3, 4, 5, 6}
A u C = {0, 1, 2, 3, 4 } u {2, 4, 6, 7}
A u C = {0, 1, 2, 3, 4, 6, 7}
(A u B)n(A u C) = {-2, 0, 1, 2, 3, 4, 5, 6}n{0, 1, 2, 3, 4, 6, 7}
(A u B) n (A u C) = {0, 1, 2, 3, 4, 6} ---------(1)
From (1) and (2), we have
A u (B n C) = (A u B) n (A u C)
Example 5 :
Let U = {1, 2, 3, 4, ......10}, A = {5, 6, 7, 9}, find A'.
Solution :
A' = U \ A
A' = {1, 2, 3, 4, ......10} \ {5, 6, 7, 9}
A' = {1, 2, 3, 4, 8, 10}
Example 6 :
Let A = {1, 3, 5, 6}, B = {0, 5, 6, 7}, find A \ B.
Solution :
A \ B = {1, 3, 5, 6} \ {0, 5, 6, 7}
A \ B = {1, 3}
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