TYPES OF SETS

In Mathematics, sets are represented as the combination of objects whose elements are fixed and cannot be changed. In other words, a set is determined as a well-defined collection of objects. These objects are also known as elements of the set. The elements present in the set cannot be repeated in the set although can be written in any order. The set is denoted by capital letters.

For example:

P = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

The components of a set in mathematics are embedded in curly brackets distributed by commas as can be seen in the above example. In this particular article, we will aim to learn about types of sets in mathematics with examples. 

Different Types of Sets

Things seem better when they are well-arranged in an ordered manner. In mathematics, characters, numbers, symbols, objects, or anything that can be grouped or are arranged and are defined as a set.

Types of sets in maths are important to learn not only to understand the theories in math but to also apply them in day-to-day life as arranging objects that belong to the alike category and keeping them in one group helps to find things easily and looks clean as well.

The different types of sets are empty set, finite set, singleton set, equivalent set, subset, power set, universal set, superset and infinite set. Let us learn about the types of sets with examples.

Null Set/Empty Set

A set that does not contain any element is termed a null set. The cardinality of the empty set is zero. The null set or the void set is expressed by the symbol ∅ and is read as phi. In roster form, ∅ is indicated by {}.

An empty set is said to be a finite set, as the number of elements/symbols in an empty set is finite, i.e., zero(0).

Empty set or null set examples:

Example 1:

P = {y : y is a leap year between 2004 and 2008}

As we can see, between 2004 and 2008, there was no leap year. Therefore, P = ϕ.

Singleton Set

A set that has only one element is termed a singleton set.

Singleton Set Examples:

P= {y : y implies neither composite nor prime}

The given set P is a singleton set as it contains one element, i.e., one.

Q = {y : y signifies a whole number that is less than one}

The set Q set includes only one element that is zero{0} hence it is an example of a singleton set.

Similarly;

A = {p : p is a whole number that is not a natural number}

There is one whole number ‘zero’ which is not a natural number, hence it is an example of a singleton set.

Set A = { 10 } is a singleton set.

Set Y = {r : r is a even prime number}

Here Y is a singleton set because there exists only one prime number that is even and it is 2.

Also, learn about Vector Algebra here.

Finite Set

A set that contains a finite number of elements is named a finite set. In other words, we can say that a set that includes no element or a definite number of elements is said to be a finite set. The empty set is also termed a finite set.

Finite set example:

Set P = {4,5,6,7,8,9,10} is a finite set, as it has a finite number of elements.

The set of different colours in the rainbow is also an example of a finite set.

Similarly;

M = {x : x ∈ M, x < 8}

Q = {3, 5, 7, 11, 13, 17, 19 …… 113} are also examples of a finite set.

Infinite Set

Exactly opposite to the finite set, the infinite set will have an infinite number of elements. If a presented set is not finite, then it will be an infinite set.

OR

A set that has an infinite number of components is named an infinite set.

Infinite set example:

A = {p : p is a whole number}

There are infinite whole numbers. Therefore, A is an infinite set.

C= {z: z is the ordinate of a position on a provided line}

There can be infinite points on a line. Therefore, C is an infinite set.

Similarly;

B = {x : x ∈ B, x > 2}

D = {x : x ∈ D, x = 3m}

Set of all prime numbers, Set of all even numbers, Set of all odd numbers are examples of an infinite set.

All infinite sets cannot be represented in roster form.

Subset

Consider A and B to be two sets. If each element of A is present in set B or we can say that if the elements of set A belong to set B, Then A is designated a subset of B and it is denoted by the notation A ⊆ B.

The symbol ‘⊆’ is applied to signify ‘is a subset of’ or ‘is included in’.

A ⊆ B; implies A is a subset of B in other words A is contained in B.

B ⊆ A; implies B is a subset A.

Every given set is a subset of itself.

Subset Examples:

Set A= {p, q, r, s, t, u}

Set B= {m, n, o, p, q, r, s, t, u}

Then we can state A ⊆ B.

Let us take another example; X = { 3, 4, 5, 6, 7, 8, 9 } and Y = { 6, 7 }. Hereabouts we can see that set Y is a subset of set X as all the components of set Y are in set X. Therefore, we can write Y ⊆ X.

Power Set 

Let A be set, then the set of all the possible subsets of A is called the power set of A and is denoted by P(A). The number of components of the power set is given by

That is  for a set A which covers  n elements, the total number of subsets that can be created is

From this, we can state that P(A) will have

elements.

Power Set Example :

For the set {x,y,z}:

The empty set {} is a subset of {x,y,z}

And these are subsets: {x}, {y} and {z}

And these are subsets: {x,y}, {x,z} and {y,z}

And {x,y,z} is actually a subset of {x,y,z} too.

Proper Subset

Consider A and B to be two sets. Then A is declared to be a proper subset of B if A is a subset of B and A is not equivalent to B. It is expressed as A ⊂ B.

OR

Any set say “P” is supposed to be a proper subset of ‘Q’ if there is at least one element in Q, which is not available in set P. That is, a proper subset is one that contains a few components of the original set.

Proper subset example:

If A = {2, 3, 4, 7, 8}

Here n(A) = 5

B = {1, 2, 3, 4, 7, 8, 10}

Here n(B) = 7

We can observe that all the elements of A are present in B but the element ‘1, 10’ of B is not available in A. Therefore, we say that A is a proper subset of B. Symbolically, this is written as A ⊂ B.

Note:

No set is a proper subset of itself.

The null set denoted by the symbol ‘∅’ is a proper subset of every set.

Improper Subset

Suppose two sets, X and Y then X is an improper subset of Y if it includes all the elements of Y.

Universal Set

This basic set is called the “Universal Set”. The universal set is normally indicated by U, and all its subsets by the letters A, B, C, etc. For example, for the set of all integers, the Universal Set can be the set of rational numbers, in human population studies, the universal set comprises all the people in the world.

OR

This is the set that is the foundation for every other set developed. Depending upon the circumstances, the universal set is chosen. It may be a finite or infinite set. All the other sets remain the subsets of the universal set.

Universal set example:

Consider if set A = {2,3,4}, set B = {4,5,6,7} and C = {6,7,8,9, 10}

Then, we will address the universal set as, U = {2,3,4,5,6,7,8,9,10}

Equal Sets

Any two sets are declared to be equal sets if and only if they are equivalent and as well as their elements are identical.

OR

Two sets P and Q are supposed to be equal if they hold the same elements. Each element of P is an element of Q and every element of Q is an element of P.

Equal sets example:

Let P{1, 2, 3, 4, 5} and Q={y : y , for 0<y<6 , y ∈ natural numbers}

Writing Q in the tabular form {1, 2, 3, 4, 5}

Sets Symbols

Set symbols are used to define the elements of a given set. The table presents some of these symbols with their meaning.

Symbols Meaning

1)  U Universal set

2) n(Y) Cardinal number of set Y

3) c ∈ P ‘c’ is an element of set P

4) a ∉ Q ‘a’ is not an element of set Q

5) ∅ Null or empty set

6) {} Denotes a set

7) X ⊆ Y Set X is a subset of set Y