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Complement of a Set
- May 31, 2020
- Category: Complement of a Set
☼ Complement of a Set : Let A be a subset of the universal set U, then the complement of A, denoted by Aٰ or is defined by :
Aٰ = = { x : x
U, x
A }.
Clearly, x
x
A.
e.g. (i) If U = { a, b, c, d, e, f } and A = { b, d, e }, then = { a, c, f }.
(ii) If U is the set of all letters in English Alphabet and A is the set of all consonants, then is the set of all vowels.
If = Upper Bound and = Lower Bound, then we get the following some important results :
1) 1)
2) A A = A 2) A
A = A
3) Aٰ Aٰ = Aٰ 3) Aٰ
Aٰ =
4) U U = U 4) U
U = U
5)
A =
5)
A = A
6)
Aٰ =
6)
Aٰ = Aٰ
7)
U =
7)
U = U
8) A Aٰ =
8) A
Aٰ = U
9) A U = A 9) A
U = U
10) Aٰ U = A 10) Aٰ
U = U
( Complement= Opposite Vertex ) U ٰ =
& ٰ
= U