What are the different types of sets?
The different types of sets are explained below with examples.
Empty Set or Null Set:
A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ and is read as phi. In roster form, ∅ is denoted by {}. An empty set is a finite set, since the number of elements in an empty set is finite, i.e., 0.
For example: (a) The set of whole numbers less than 0.
(b) Clearly there is no whole number less than 0.
Therefore, it is an empty set.
(c) N = {x : x ∈ N, 3 < x < 4}
• Let A = {x : 2 < x < 3, x is a natural number}
Here A is an empty set because there is no natural number between
2 and 3.
• Let B = {x : x is a composite number less than 4}.
Here B is an empty set because there is no composite number less than 4.
Note:
∅ ≠ {0} ∴ has no element.
{0} is a set which has one element 0.
The cardinal number of an empty set, i.e., n(∅) = 0
Singleton Set:
A set which contains only one element is called a singleton set.
For example:
• A = {x : x is neither prime nor composite}
It is a singleton set containing one element, i.e., 1.
• B = {x : x is a whole number, x < 1}
This set contains only one element 0 and is a singleton set.
• Let A = {x : x ∈ N and x² = 4}
Here A is a singleton set because there is only one element 2 whose square is 4.
• Let B = {x : x is a even prime number}
Here B is a singleton set because there is only one prime number which is even, i.e., 2.
Finite Set:
A set which contains a definite number of elements is called a finite set. Empty set is also called a finite set.
For example:
• The set of all colors in the rainbow.
• N = {x : x ∈ N, x < 7}
• P = {2, 3, 5, 7, 11, 13, 17, ...... 97}
Infinite Set:
The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set.
For example:
• Set of all points in a plane
• A = {x : x ∈ N, x > 1}
• Set of all prime numbers
• B = {x : x ∈ W, x = 2n}
Note:
All infinite sets cannot be expressed in roster form.
For example:
The set of real numbers since the elements of this set do not follow any particular pattern.
Cardinal Number of a Set:
The number of distinct elements in a given set A is called the cardinal number of A. It is denoted by n(A).
For example:
• A {x : x ∈ N, x < 5}
A = {1, 2, 3, 4}
Therefore, n(A) = 4
• B = set of letters in the word ALGEBRA
B = {A, L, G, E, B, R}
Therefore, n(B) = 6
Equivalent Sets:
Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.
For example:
A = {1, 2, 3} Here n(A) = 3
B = {p, q, r} Here n(B) = 3
Therefore, A ↔ B
Equal sets:
Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A.
For example:
A = {p, q, r, s}
B = {p, s, r, q}
Therefore, A = B
The various types of sets and their definitions are explained above with the help of examples.
● Set Theory
● Sets
● Subset
8th Grade Math Practice
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