Types of Sets

We know that a set is a well–defined collection of objects. There are different types of sets depending on the objects and their characteristics. Some of these are explained below. Let us understand the various types of sets along with illustrations.

Types of sets are classified according to the number of elements they have. Sets are the collection of elements of the same type. For example, a set of prime numbers, natural numbers, etc. There are various types of sets such as unit sets, finite and infinite sets, null sets, equal and unequal sets, etc. Let us learn more about the various forms of sets in detail.

Singleton Sets or Unit Sets

A set that has only one element is called a singleton set. It is also known as a unit set because it has only one element. Example, Set A = { k | k is an integer between 5 and 7} which is A = {6}.

Finite Sets

As the name implies, a set with a finite or exact countable number of elements is called a finite set. If the set is non-empty, it is called a non-empty finite set. Some examples of finite sets are: For example, Set B = {k | k is a even number less than 20}, which is B = {2,4,6,8,10,12,14,16,18}. Let us consider one more illustration, Set A = {x : x is a day in a week}; Set A will have 7 elements.

Infinite Sets

A set with an infinite number of elements is called an infinite set. In other words, if a given set is not finite, then it will be an infinite set. For example, A = {x : x is a real number}; there are infinite real numbers. Hence, here A is an infinite set. Let us consider one more example, Set B = {z: z is the coordinate of a point on a straight line}; there are infinite points on a straight line. So, here B is an example of an infinite set. Another example could be Set C = {Multiples of 3}. Here we can have infinite multiples of 3.

Empty or Null Sets

A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as 'phi'. Example: Set X = {}.

Equal Sets

If two sets have the same elements in them, then they are called equal sets. Example: A = {1,3,2} and B = {1,2,3}. Here, set A and set B are equal sets. This can be represented as A = B.

Unequal Sets

If two sets have at least one element that is different, then they are unequal sets.Example: X = {4, 5, 6} and Y = {2,3,4}. Here, set X and set Y are unequal sets. This can be represented as X ≠ Y.

Equivalent Sets

Two sets are said to be equivalent sets when they have the same number of elements, though the elements are different. Example: A = {7, 8, 9, 10} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)

Overlapping Sets

Two sets are said to be overlapping if at least one element from set A is present in set B. Example: A = {4,5,6} B = {4,9,10}. Here, element 4 is present in set A as well as in set B. Therefore, A and B are overlapping sets.

Disjoint Sets

Two sets are disjoint sets if there are no common elements in both sets. Example: A = {1,2,3,4} B = {7,8,9,10}. Here, set A and set B are disjoint sets.

Subset and Superset

For two sets A and B, if every element in set A is present in set B, then set A is a subset of set B(A ⊆ B) and B is the superset of set A(B ⊇ A).
Example: A = {1,2,3} B = {1,2,3,4,5,6}
A ⊆ B, since all the elements in set A are present in set B.
B ⊇ A denotes that set B is the superset of set A.

Universal Set

A universal set is the collection of all the elements in regard to a particular subject. The set notation used to represent a universal set is the letter 'U'. Example: Let U = {The list of all road transport vehicles}. Here, a set of cars is a subset for this universal set, the set of cycles, trains are all subsets of this universal set.

Power Sets

Power set is the set of all subsets that a set could contain. Example: Set A = {1,2,3}. Power set of A is = {{∅}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

☛ Related Topics:

Check out some interesting topics related to types of sets.

• Operations on Sets
• Roster Notation
• Intersection of Sets
• Set Builder Notation

What are the Different Types of Sets in Mathematics?

There are various types of sets like – finite and infinite sets, equal and equivalent sets, a null set, etc. Further, there are a subset and proper subset, power set, universal set, disjoint sets, etc depending on the characteristics of the sets.

What are Finite and Infinite Types of Sets?

Any set that is empty or consists of a definite and countable number of elements is referred to as a finite set. Whereas, sets with uncountable or indefinite numbers of elements are called infinite sets.

What is a Universal Type of Set?

A universal set consists of all the elements of a problem under consideration. We generally represent it by the letter U. For instance, the set of real numbers is a universal set for all-natural, whole, odd, even, rational in addition to irrational numbers.

What are Equal and Unequal Types of Sets?

Two sets A and B are said to be equal if they have exactly similar elements. Here the elements can be irrespective of the order of appearance in the set. Equal sets are represented as A = B. Otherwise, the sets are referred to as unequal sets, which are represented as A ≠ B. For example, if A = {1, 2, 5} and D = {1, 5, 2} then both of these sets are equal.

Is Empty Set a Type of Finite Set?

An empty set is a type finite set as it contains no elements. The number of elements in an empty set is definite, that is, zero, therefore, it is a finite set.

What are the Examples of Finite and Infinite Types of Sets?

Some common examples of finite and infinite sets are listed below:

• Let S be a set of natural numbers, then S is an infinite set.
• Let W be a set of weeks in a year, then W is a finite set.