This 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, (i) , (ii) are null sets. A null set is also called an empty or void set and is denoted by the Greek letter.
This 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 .
A set consisting of only one element is called a singleton set. For example , are singleton 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 must always 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 .
A set is said to be a subset of a set if every member of is also a member of . Using the inclusion symbol or , we write this fact as and read it 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.
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.
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.