# 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**.