Definition and List of Different Types of Sets with Examples and Properties
In the field of Mathematics, sets can be defined as the collection of objects whose elements are fixed and cannot be changed. You can say that a set is a well defined collection of objects. The elements cannot be repeated in a set but can be written in any order. The set is always represented by capital letters.
What are the types of Sets?
There are primarily 8 types of sets that are used in Mathematics, they are:
Empty Sets - The set, which has no elements, is also called a Null set or Void set. It is denoted by {}.
Singleton Sets - The set which has just one element is named a singleton set.
Finite and Infinite Sets - A set that has a finite number of elements is known as a finite set, whereas the infinite set is the set whose elements can't be estimated, but it has some figure or number that is adequate enough to evaluate that set.
Equal Sets - If every element of set A is also the element of set B and if every element of set A is also the elements of set A are called equal sets. This implies that the elements of both the sets i.e. set A and set B are equal.
Subsets - A set P is said to be a subset of set U if the elements of set U belong to set P. In other words, it can be said that each and every element present in the set P is also present in set U.
Power Sets - The set of all subsets is known as power sets.
Universal Sets - A set that contains all the elements of other sets is called a universal set.
Disjoint Sets - If two sets X and Y do not have any common elements, and their intersection results in zero (0), then set X and Y are called disjoint sets.
Union, Intersection,Difference and Complement of Sets -
Union of Sets -
The union of two sets consists of all their elements. It is denoted by (⋃).
For example: Set A = {2,3,7} and set B = { 4,5,8}
Then the union of set A and set B will be;
B ⋃ B = {2,3,7,4,5,8}
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Intersection of Sets -
The set of all elements, which are common to all the given sets, gives an intersection of sets. It is denoted by ⋂.
For Example: set A = {2,3,7} and set B = {2,4,9}
So, A ⋂ B = {2}
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Difference of Sets -
The difference between set S and set T is such that it has only those elements which are in the set S and not in the set T. S – T = {p : p ∊ S and p ∉ T}
Similarly, T – S = {p: p ∊ T and p ∉ S}
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Complement of a Set
Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U - A),is defined as:
A’ = {x U : x A}
Clearly, x A’ x A
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Every set has a complement of sets. Also, for a universal set, the empty set is known as the complement of the universal set. The empty set contains no elements of the subset and is also known as Null Set, which is denoted by {Ø} or {}.
Questions to be Solved:
Solved Examples
1.If set A = {a, b, c, d} and B = {b, c, e, f} then, find A-B.
Answer: Let’s find the difference of the two sets,
A – B = {a, d} and B – A = {e, f}
2.Let X = {David, Jhon, Misha} be the set of students of Class XI, who are in the school hockey team. Let Y = {Zoya, Rahul, Riya} be the set of students from Class XI who are in the school football team. Find X U Y and interpret the set.
Solution:
(U Union - Combination of two sets)
Given X = {David, Jhon, Zoya}
Y = {Zoya, Rahul, Riya}
Common elements (Zoya) should be taken once
X U Y = {David, Jhon, Zoya, Rahul, Riya}.
This union set is equal to the set of students from Class eleven who are present in the hockey team or in the football team or in both of the teams.
FAQs on Types of Sets in Mathematics Explained Clearly
1. What are the different types of sets in mathematics?
The main types of sets in mathematics are empty set, finite set, infinite set, singleton set, equal set, equivalent set, subset, universal set, disjoint set, and power set.
- Empty set (∅): A set with no elements.
- Finite set: A set with a limited number of elements.
- Infinite set: A set with unlimited elements.
- Singleton set: A set with exactly one element.
- Equal sets: Sets having exactly the same elements.
- Equivalent sets: Sets having the same number of elements.
- Subset: A set whose elements are all contained in another set.
- Universal set: The set containing all elements under discussion.
- Disjoint sets: Sets with no common elements.
- Power set: The set of all subsets of a given set.
2. What is an empty set in set theory?
An empty set is a set that contains no elements and is denoted by ∅ or { }. For example, the set of natural numbers less than 0 is ∅ because no natural number is less than 0. The empty set is also called the null set and has cardinality 0.
3. What is the difference between finite and infinite sets?
A finite set has a limited number of elements, while an infinite set has infinitely many elements.
- Example of finite set: A = {1, 2, 3, 4} (number of elements = 4).
- Example of infinite set: N = {1, 2, 3, 4, ...} (natural numbers continue endlessly).
4. What is a singleton set with an example?
A singleton set is a set that contains exactly one element. For example, A = {5} is a singleton set because it has only one member. The cardinality of a singleton set is always 1.
5. What are equal sets in mathematics?
Two sets are called equal sets if they contain exactly the same elements, regardless of order. If A = {1, 2, 3} and B = {3, 2, 1}, then A = B because every element of A is in B and vice versa. Equal sets satisfy the condition A = B.
6. What is the difference between equal sets and equivalent sets?
The difference is that equal sets have the same elements, while equivalent sets have the same number of elements.
- Equal sets example: A = {1, 2, 3}, B = {3, 2, 1} → A = B.
- Equivalent sets example: A = {1, 2, 3}, C = {a, b, c} → Both have 3 elements, so they are equivalent.
7. What is a subset in set theory?
A subset is a set whose every element belongs to another set and is denoted by ⊆. If A = {1, 2} and B = {1, 2, 3}, then A ⊆ B because all elements of A are in B. If A is contained in B but A ≠ B, then A is called a proper subset, written as ⊂.
8. What is a universal set with an example?
A universal set is the set that contains all elements under consideration in a particular context and is usually denoted by U. For example, if U = {1, 2, 3, 4, 5} and A = {1, 2}, then A is a subset of U. The universal set depends on the problem being discussed.
9. What are disjoint sets?
Two sets are called disjoint sets if they have no common elements, meaning their intersection is the empty set. If A = {1, 2} and B = {3, 4}, then A ∩ B = ∅. Disjoint sets satisfy the condition A ∩ B = ∅.
10. What is a power set and how do you find it?
The power set of a set is the set of all possible subsets of that set and is denoted by P(A). If A = {1, 2}, then
- Subsets are: ∅, {1}, {2}, {1, 2}
