# Some Specified Sets

__Null Set__

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

__Universal Set__

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 $$U$$.

__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 must always 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 the inclusion symbol $$ \supset $$ or $$ \subset $$, we write this fact as $$B \subset A$$ and read it as “$$A$$ contains (or includes) $$B$$” or “$$B$$ is contained (or included) in $$A$$.” 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**.