# 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) 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.