# 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.