Set is defined as a collection of objects. There are many types of sets in mathematics which are given below.
Types of Sets
1). Empty Set
A set is said to be empty or null or void set if it has no element and it is denoted by \(\phi\).
for example : The set A is given by A = [ x : x is an even prime number greater than 2 ] is an empty set because 2 is the only even prime number.
2). Singleton Set
A set consisting of a single element is called a singleton set.
for example : The set {5} is a singleton set.
3). Finite Set
A set is called finite set if it is either void set or its element can be listed by the natural numbers 1, 2, 3 ….. n for any natural number n.
for example : The set of even natural numbers less than 100.
4). Infinite Set
A set whose elements cannot be listed by the natural numbers 1, 2, 3 ….. n for any natural number n is called infinite set.
for example : The set of all points in a plane.
5). Equivalent Sets
Two finite sets A and B are said to be equivalent if their cardinal numbers are same i.e. n(A) = n(B).
for example : If A = { 1, 2 } and B = { 3, 4 }, both are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A| = |B| = 2.
6). Equal Sets
Two sets A and B are said to be equal if every element of A is a member of B, and every element of B is a member of A.
If sets A and B are equal, we write A = B and A \(\ne\) B when A and B are not equal.
for example : If A = { 1, 2, 5, 6 } and B = { 5, 6, 2, 1 }, then A = B, because each element of A is an element of B and vice versa.
7). Subsets
Let A and B be two sets. If every element of A is a element of B, then A is called a subset of B.
If A is a subset of B, we write A \(\subset\) B, which is read as “A is a subset of B”.
for example : If A = { 1 } and B = { 3, 2, 1 }, then A \(\subset\) B, because every element of A is an element of B.
8). Universal Set
A set that contains all sets in a given context is called universal set.
for example : when we are using sets containing natural numbers, then N is the universal set.
9). Power Set
Let A be a set. Then the collection or family of all subsets of A is called the power set of A and is denoted by P(A).
Since the empty set and set A itself are subsets of A and are therefore elements of P(A). Thus the power set of a given set is always non-empty.
for example : Let A = {1, 2}. Then the subsets of A are :
\(\phi\), {1}, {2}, {1, 2}
Hope you learnt types of sets in mathematics , learn more concepts of sets and practice more questions to get ahead in the competition. Good luck!
