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A - B = {x: x belongs to A and x does not belong to B}
Complement:
Complement is always taken with respect to a universal set
U = Universal Set
Universe is the set of all alphabets.
U = {a, b, c, d, ..., x, y, z}
A = {a, e, i, o, u}
A' = U - A
A union A' = U
A' = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}
Assuming a universe U and a set A where A is a subset of U
A' = {x: x belongs to U and x does not belong to A}
A' = U - A
U = {a, b, c, d, e, f, g}
A = {a, b, c, d}
A' = {e, f, g}
U = Set of natural numbers
A = Set of even natural numbers
A' = Set of odd natural numbers
U = {a, b, c, d, e, f, g}
A = {a, b, c, d}
A' = {e, f, g}
(A')' = ({e, f, g})'
= {a, b, c, d}
(A')' = A
Problem:
U = {1, 2, 3, 4, ..., 8, 9, 10}
A = {1, 2, 3, 4, 5}
B = {2, 4, 6, 8, 10}
C = {2, 3, 5, 7}
Complements:
A' = {6, 7, 8, 9, 10}
B' = {1, 3, 5, 7, 9}
C' = {1, 4, 6, 8, 9, 10}
A union B = {1, 2, 3, 4, 5, 6, 8, 10}
B union C = {2, 3, 4, 5, 6, 7, 8, 10}
A union C = {1, 2, 3, 4, 5, 7}
(A union B)' = {7, 9}
A' intersection B' = {7, 9}
(B union C)' = {1, 9}
B' intersection C' = {1, 9}
(A union C)' = {6, 8, 9, 10}
A' intersection C' = {6, 8, 9, 10}
(A union B)' = A' intersection B'
Law 2:
A intersection B = {2, 4}
B intersection C = {2}
A intersection C = {2, 3, 5}
(A intersection B)' = {1, 3, 5, 6, 7, 8, 9, 10}
(B intersection C)' = {1, 3, 4, 5, 6, 7, 8, 9, 10}
(A intersection C)' = {1, 4, 6, 7, 8, 9, 10}
A' union B' = {1, 3, 5, 6, 7, 8, 9, 10}
B' union C' = {1, 3, 4, 5, 6, 7, 8, 9, 10}
A' union C' = {1, 4, 6, 7, 8, 9, 10}
(A intersection B)' = A' union B'
De-Morgan's Laws:
(A union B)' = A' intersection B'
(A intersection B)' = A' union B'
Properties of complement:
A union A' = U
A intersection A' = phi
(A union B)' = A' intersection B'
(A intersection B)' = A' union B'
(A')' = A
(phi)' = U
U' = phi
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