Well, there are different types of sets in mathematics that are based on the number of elements, pairs, relationships between two sets.
In our daily life, when we arrange things in a well-mannered way, then these look better to us. Likewise in mathematics, when we arrange characters, symbols, numbers, and objects in different groups then these are categorized as different types of sets of mathematics.
The different types of sets in mathematics are a finite set, infinite set, empty set, singleton set, equal sets, equivalent and non-equivalent sets, subset, proper subset, improper subset, and power set. Now, we are going to learn about these types of sets in detail:
Empty Set:
A set that does not possess any element is called an empty set. The empty set is stated by a symbol ∅ and is read as phi. It is represented by {}.
As an empty set has 0 elements, therefore, it is also called a finite set.
Example:
P = The set of Odd numbers is less than 1.
Here, we know that the set of Odd numbers is {1, 3, 5,....}. There doesn’t exist any odd number less than 1. So, P is an empty set. It is denoted by
P = {}
Q = The set of rectangles with 5 sides
Since there don’t exist rectangles with 5 sides. Therefore, Q is an empty set.
Q = {}
W = The set of months with 40 days
As you know that every month contains 30 or 31 days. That is why,
W = {}
Empty set example in real life
2. Finite Sets:
A set that contains no element or a finite number of elements is called a finite set. Furthermore, a finite set has a limited or definite number of elements.
An empty set is also named a finite set.
Example:
R = A set of prime numbers less than 11.
R = {2, 3, 5, 7, 11}
Here, elements in R are countable and finite. So, R is a finite set.
S = Set of planets in a solar system
S = { Earth, Jupiter, Saturn, Mercury, Venus, Mars, Uranus, Neptune}
The planets of the solar system are countable so S is a finite set.
T = set of colors in a rainbow
T = { Red, orange, yellow, green, blue, indigo, violet }
T is also a finite set.
Example of finite set
3. Infinite Sets:
A set that contains unlimited elements or an infinite number of elements is called an infinite set. In addition, infinite sets are exactly the opposite of finite sets.
Example:
L = set of leaves of trees
You know that leaves of trees are unlimited and can’t be countable so L is an infinite set.
Z = set of positive integers
Here, Z = { 0, 1, 2, 3,...}
The elements of Z are unlimited and thus Z is an infinite set.
X = set of grains of sand in a seashore
As you know the grains of sand are very tiny particles and can’t
Be countable. So X is an infinite set.
Examples of infinite set in daily life
4. Singleton Sets:
A set that accommodates only one element is called a singleton set.
Example:
H = set of counting numbers between 3 and 5
Here number between 3 and 5 is 4 which is only a single element, so, H is called a singleton set.
F = Set of numbers that are both composite and prime
You know that only 1 is a number that is a composite as well as prime. So, F is a singleton set.
R = set of natural numbers less than 2
Since there exists only one natural number less than 2 that is 1. So, R is a singleton set.
Examples of singleton set in daily life
5. Equal Sets:
A pair of sets are said to be equal if and only if they have an equal number of elements as well as the same elements.
In other words, if M and N are any two sets. Then, each element of M is an element of N, and every element of N is also an element of M.
Equal sets are denoted by ‘=’.
Example:
Y = {o, r, a, n, g, e}
Z = {a, e.g, o, r, n}
6. Equivalent Sets:
A pair of sets are said to be equivalent if they have only the same number of elements.
In other words, If M and N are any two sets and if M has 4 elements then N must have 4 elements. Thus, we can say that M and N are equivalent sets.
Equivalent sets are denoted by ‘↔’.
Examples of Equivalent Sets:
7. Non-Equivalent Sets:
A pair of sets are said to be non-equivalent if they do have not the same number of elements. Furthermore, Equivalent and non-equivalent sets are opposite of each other.
In other words, M and N are said to be non-equivalent if M has 3 elements and N has 4 elements.
Non-Equivalent sets are symbolized by ‘⇎’.
Examples of Non-Equivalent sets:
Practice Questions For Types of Sets in Mathematics
Question#1:
Check out whether the following sets are empty or not?
(i) The set of prime numbers less than 2
(ii) The set of the English alphabet between x and y
(iii) The set of years with 400 days.
(iv) The set of 40 feet tall girls.
(v) The set of integers which are called prime numbers
Question#2:
Indicate the finite and infinite sets.
A = {p, e, a, r}
B = The set of Real numbers
C = {1, 2, 3,...,365}
D = {potato, onion, carrot, ladyfinger}
E = {4,8,12,...}
F = {a, b, c,...,z}
G = {1/3, 2/5, 3/7,...}
Question#3:
Tell whether the following pairs of sets are equivalent or nonequivalent.
(i) A = {x, y}, B = {1,2,7}
(ii) A = {pen, notebook, ruler, book}, B = {5,6,7,8}
(iii) A = {-1, 0, 1, 2}, B = {@, #, $,%}
(iv) A = {0,1,2,3,4}, B = {1,2, 4, 0, 3}
(v) A = {mango, apple, banana, orange}, B = {cat, dog, cow, hen}
Question#4
Are the given pairs of sets are equal sets?
(i) L = {a, f, g, h, e}, M = {g, f, e, h, a}
(ii) W = {11, 12, 13, 14}, X = {1+10, 11+1, 11+2, 11+3}
(iii) P = {3, 2, 1, 4,0}, Q = {6-2, 5-1, 4-2, 3-1, 2-2}
(iv) S = The set of even numbers less than 7, T = {4, 0, 2, 6}
(v) Y = Set of 4 girls name start with ‘s’. Z = { sana, smreen, sidra, sofia}