Null Set: It is a set which has no member. This set may be specified by defining a property for the members such that no object may satisfy this property. For example, (1) , (ii) are null sets. A null set is also called empty or void set and is denoted by the Greek letter.
Universal Set: It is the largest set which contains all objects likely to be considered during some specified mathematical treatment. We denote such a set by the letter .
Singleton Set: A set consisting of only one element is called a singleton set. For example , are singleton sets.
Equal Sets: Two sets and are said to be equal if each member of is a member of and conversely. In such a case we write and say that the two sets and are equal.
To prove the equality of two sets we have always to show that each member of one set is a member of the other set. If two sets and are unequal then we write .
It is interesting to know that there can be only one null set although we may define it differently. To prove this, suppose and are different null sets. Since , are null sets, by definition . Thus every member of if there is any, is
a member of. Similarly, , concludes that every member of , if there
is any, is a member of . Hence ,, cannot be different. This establishes the uniqueness of the null set. A null set is a finite set whose number of members is .
Subsets: A set is said to be a subset of a set if every member of is also a member of .Using inclusion symbol or, we write this fact as and read as “ contains (or includes) ” or “ is contained (or included) in” Evidently, If and , then is called a superset of , and is called a proper subset of . When is not a subset of we write . We have iff or .
By convention, as it is also implied from the definition, the null set is regarded as a subset of every set.
Comparable Sets: Two sets are said to be comparable if one contains the other. If none contains the other than the two sets are said to be incomparable. For example, sets are comparable, and and are incomparable sets.
Disjoint Sets: Two sets are said to be disjoint if they have no common member. For example, and are disjoint sets.
Family of Sets: A set of sets is called a family (or class) of sets. For example, if represents a set for each contained in some set then the set is a family of sets. The set used above is called an index set.