# Some Specified Sets

__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**.