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Complement of a Set

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☼ Complement of a Set : Let A be a subset of the universal set U, then the complement of A, denoted by Aٰ or $A^{c}$ is defined by :

Aٰ = $A^{c}$ = { x : x $\in$ U, x $\notin$ A }.

Clearly, x $\in$ $A^{c}$ $\large \Leftrightarrow$ x $\notin$ A.

e.g. (i) If U = { a, b, c, d, e, f } and A = { b, d, e }, then $A^{c}$ = { a, c, f }.

(ii) If U is the set of all letters in English Alphabet and A is the set of all consonants, then $A^{c}$ is the set of all vowels.

If = Upper Bound and = Lower Bound, then we get the following some important results :

1) $\emptyset \cap \emptyset = \emptyset$ 1) $\emptyset \cup \emptyset = \emptyset$

2) A $\cap$ A = A 2) A $\cap$ A = A

3) Aٰ $\cap$ Aٰ = Aٰ 3) Aٰ $\cup$ Aٰ = $a^{c}$

4) U $\cap$ U = U 4) U $\cup$ U = U

5) $\emptyset$ $\cap$ A = $\emptyset$ 5) $\emptyset$ $\cup$ A = A

6) $\emptyset$ $\cap$ Aٰ = $\emptyset$ 6) $\emptyset$ $\cup$ Aٰ = Aٰ

7) $\emptyset$ $\cap$ U = $\emptyset$ 7) $\emptyset$ $\cup$ U = U

8) A $\cap$ Aٰ = $\emptyset$ 8) A $\cup$ Aٰ = U

9) A $\cap$ U = A 9) A $\cup$ U = U

10) Aٰ $\cap$ U = A 10) Aٰ $\cup$ U = U

( Complement= Opposite Vertex ) U ٰ = $\emptyset$ & ٰ $\emptyset`$ = U