Types of Set
1.Finite Set
When the number of elements of a set is finite or countable, or its cardinality is a natural number(∈ N), it forms a Finite set.
Take, for example, A={1,4,9,16,25} or B={a,b,c,d,e,f}. Both A and B have a definite number of elements. A has 5 elements and B has 6 elements. Hence, they are finite sets.
Here are a few set conditions that are always finite.
- A subset of Finite set
- The union of two finite sets
- The power set of a finite set
2.Infinite Set
A set whose number of elements(cardinality) can not be counted or a set that is not finite is called an Infinite set. The cardinality of this set cannot be determined. Infinite sets are also called uncountable sets. Some examples of infinite sets are the set of all points on a line, the set of all whole numbers: W= {0, 1, 2, 3, 4,…}, set of multiples of 10.
Here are a few set conditions that are always infinite.
- The union of two infinite sets
- The power set of an infinite set
- The superset of an infinite set
3.Singleton Set
A set whose cardinality is one or has only one element is known as a Singleton set. Since it has only one element, it is also called a unit set.
For example, A = {5} or B={x}
4.Empty Set
If the cardinality of a set is zero, or there are no elements in a set, it is an empty set. It is represented as { }. The cardinality is zero, which is a definite number, so it is a finite set. It is also called a null or void set, denoted by ∅.
5.Equal Set
Two sets are said to be equal if their cardinalities and all their elements are the same, irrespective of the order.
For example, A={x,y,z} and B={z,x,y}. Sets A and B are said to be equal because they have three elements each, and all their elements are the same.
6.Equivalent Set
Two sets are equivalent if their cardinalities,i.e., the number of elements, are the same.
For example, A={x,y,z} and B={a,b,c}. A and B are equivalent sets because they have three elements each.
7.Subset
A set X is a subset of set Y if all the elements of set X are in set Y, i.e., every element of set X is also in set Y. The symbol ⊆ stands for “is contained in” or “is a subset of.” Thus, if X is contained in Y, we write
X ⊆ Y. Here, Y is the superset of X.
Take, for example, X={a,b,c,d,e} and Y=All letters of the English alphabet. So, X⊆Y since the elements of X are also elements of Y. Also, Y⊆X means Y contains X or is a superset of X.
Proper Subset
A set X is a proper subset of set Y if X is a subset of Y and X ≠ Y. Precisely, if each element of X is an element of Y and at least one element of X is not an element of Y, then X is said to be a proper subset of Y. The symbol ⊂ represents “is a proper subset of.” The null set(empty) is a proper subset of every set except itself.
Improper Subset
A set X is an improper subset of set Y if X is a subset of Y and X=Y. Every set is an improper subset of itself.
8.Universal Set
A set comprising all the elements of all the sets is the Universal set for those sets. It is denoted by U or S.
For example :
Consider the following sets, X = {a, b, c, d, e} ; Y = {x, y, z} and U =All letters of the English Alphabet. Here, U is the universal set for sets X and Y since U contains all elements of X and Y.
FAQs
1.What is the difference between Equivalent Sets and Equal Sets?
Two sets are equivalent if their cardinality is the same, but all the elements may not be the same, whereas two sets are equal if their cardinality and all the elements are the same. Equal sets are equivalent, but equivalent sets may or may not be equal.
2.How is an empty set a finite set?
An empty set has no elements in it, i.e., its cardinality is zero. Since zero is a definite number therefore an empty set is a finite set.
3.What are Disjoint Sets?
If two sets do not have any common element they are called disjoint sets. Intersection of these sets is ∅ and n(A ∪ B) = n(A) + n(B).
Key Takeaways
In this article, we have extensively discussed Introduction to sets, its mathematical definition, and the types of sets. We hope that this blog has helped you enhance your knowledge, and if you wish to learn more, check out our playlist Basic Mathematics.
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