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) \left\{ {x:x \ne x} \right\}, (ii) \left\{ {x:x \in \mathbb{R}{\text{ and  }}{x^2} = - 1} \right\} are null sets. A null set is also called empty or void set and is denoted by the Greek letter\phi .

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 U.

Singleton Set: A set consisting of only one element is called a singleton set. For example \left\{  0 \right\}, \left\{ {11} \right\} are singleton sets.

Equal Sets: Two sets A and B are said to be equal if each member of A is a member of B and conversely. In such a case we write A = B and say that the two sets A and B 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 A and B are unequal then we write A \ne B.

It is interesting to know that there can be only one null set although we may define it differently. To prove this, suppose {\phi _1} and {\phi _2} are different null sets. Since {\phi _1},{\phi  _2} are null sets, by definition x  \notin {\phi _1} \Rightarrow x \notin {\phi _2}. Thus every member of {\phi _2} if there is any, is
a member of{\phi _1}. Similarly, x \notin {\phi _2} \Rightarrow x \notin {\phi _1}, concludes that every member of {\phi _1}, if there
is any, is a member of {\phi _2}. Hence {\phi _1},{\phi  _2}, cannot be different. This establishes the uniqueness of the null set. A null set is a finite set whose number of members is 0.

Subsets: A set B is said to be a subset of a set A if every member of B is also a member of A.Using inclusion symbol  \supset or \subset , we write this fact as B \subset A and read as “A contains (or includes) B or “B is contained (or included) inA Evidently, A \supset A,A  \subset A If A \ne B and A \supset B, then A is called a superset of B, and B is called a proper subset of A. When B is not a subset of A we write B \not\subset A. We have A = B iff  A \supset B or B \supset A.

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 \left\{ {1,2,3}  \right\},\left\{ {3,4,2,1} \right\} are comparable, and \left\{ {1,2,3} \right\} and \left\{ {1,4,5} \right\} are incomparable sets.

Disjoint Sets: Two sets are said to be disjoint if they have no common member. For example, \left\{  {1,2,3} \right\} and \left\{ {4,5}  \right\} are disjoint sets.

Family of Sets: A set of sets is called a family (or class) of sets. For example, if {A_i} represents a set for each i contained in some set I then the set \left\{ {{A_i}:i \in I} \right\} is a family of sets. The set I used above is called an index set.