In the theory of set, there are different types of sets. Every procedure in set theory based on sets. Set should be a collection of individual terms in domain. The universal set has each and every element of domain. We have explained different types of sets.

### Different Types of Sets

Following are the different types of sets in set theory:

• Empty set
• Singleton set
• Finite and Infinite set
• Union of sets
• Intersection of sets
• Difference of sets
• Subset of a set
• Disjoint sets
• Equality of two sets

### Empty set:

A set that has no element is said to be as an Empty set. We can also say it as Null set and Void set. The number of element in set A is represented as n(A). The empty set is symbolized as Φ. Thus, n(Φ) = 0. The cardinality of an empty set is zero since it has no element.

### Singleton Set

A set that has 1 and only 1 element is said to be as Singleton set.  We can also name it as unit set. The cardinality of singleton is 1. If Z is a singleton, then we can express it as
Z = {x : x = Z}

Example: Set Z = {5} is a singleton set.

### Finite and Infinite Set

A set that has predetermined number of elements or finite number of elements are called Finite set. Like {11 ,12, 13, 14, 15, 16} is finite set whose cardinality is 16, since it has 16 elements.

Else, it is called as infinite set. It may be uncountable or countable. The union of some infinite sets are infinite and the power set of any infinite set is infinite.

Examples:

1. Set of all the weeks in a month is a finite set.
2. Set of all integers is infinite set.

### Union of sets:

Union of 2 or else most numbers of sets could be the set of all elements that belongs to every element of all sets. In the union set of 2 sets, each element is written only once even if they belong to both the sets. This is represented as ‘. If we have sets Y and Z, then the union of these two is Y U Z and called as Y union Z.

Mathematically, we can denote it as Y U Z = {x : x Y or x Z}

The union of two sets is always commutative i.e. Y U Z = Z U Y.

Example: Y = {1,2,3}

Z = {1,4,5}

Y Z = {1,2,3,4,5}

### Intersection of Sets:

A set of elements that are common in both the sets. Intersection is similar to grouping up the common elements. The symbol should be symbolized as ‘∩’. If X and Y are two sets, then the intersection is represented as X Y and called as X intersection Y and mathematically, we can write it as
XY={x:xXxY}

Example: X = {1,2,3,4,5}
Y = {2,3,7}
X
Y = {2,3}

### Difference of Sets

The difference of set X to Y should be denoted as X - Y. That is, the set of element that are in set X not in set Y is
X - Y = {x: x X and x Y}

And, Y - X is the set of all elements of the set B which are in B but not in A i.e.
Y - X = {x: x Y and x X}.

Example:

If A = {1,2,3,4,5} and B = {2,4,6,7,8}, then
A - B = {1,3,5} and B - A = {6,7,8}

### Subset of a Set

In set theory, a set X is the subset of any set Y, if the set X is contained in set Y. It means, all the elements of the set X also belong to the set Y. It is denoted as ' or X Y.

Example:

X = {1,2,3,4,5}
Y = {1,2,3,4,5,7,8}
Here, X is said to be the subset of Y.

### Disjoint Sets

If two sets X and Y should not have common elements or if the intersection of any 2 sets X and Y is the empty set, then these sets are called disjoint sets i.e. X Y = ϕ. That means, when this condition n (X ∩ Y) = 0 is true, then the sets are disjoint sets.

Example:
X = {1,2,3}
Y = {4,5}
n (A ∩ B) = 0.
Therefore, these sets X and Y are disjoint sets.

### Equality of Two Sets

Two sets are called equal or identical to each other, if they contain the same elements. When the sets X and Y is said to be equal, if X Y and Y X, then we will write as X = Y.

Examples:

1. If X = {1,2,3} and Y = {1,2,3}, then X = Y.
2. Let P = {a, e, i, o, u} and Q = {a, e, i, o, u, v}, then P ≠ Q, since set Q has element v as the additional element.

March 09, 2018