The set with no elements is called the empty (or null) set and is denoted f Two sets X and Y are equal iff X and Y have the same elements.

i.e. X = Y iff whenever x Î X then x Î Y and whenever x Î Y ,

then x Î X.

If X and Y are sets and if every element of X is an element of Y, then we say that X is a subset of Y, denoted as X Í Y. If A and Y are not equal, then X is proper subset of Y.

The set of all subsets of a set X , is called the power set of X, denoted as P(X). If |X| = n , then |P(X)| = 2 ⁿ

Set Operations :

Union : X È Y = {x | x Î X or x Î Y }

Intersection : X Ç Y = {x | x Î X and x Î Y }

X and Y are disjoint if X Ç Y = f

Difference : X - Y = {x | x Î X and x Ï Y }

Sometimes we are dealing with sets all of which are subsets of a set U. The set U is called the universal set or a universe. The set U must be explicitly given or inferred from the context. Given the universal set U and a subset X of U, the set U - X is called the complement of X and is written as X^c .(sometimes, as X' ).

|A È B | = |A| + |B| - |A Ç B |

Theorem 2.1.10 page 59 :

Let U be a universal set and let A,B, and C be subset of U. The following properties hold: