Sets in mathematics, are simply a collection of distinct objects forming a group. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. Every item in the set is called an element of the set. Curly brackets are used while writing a set. A very simple example of a set would be like this. Set A = {1,2,3,4,5}. There are various notations to represent elements of a set. Sets are usually represented using a roster form or a set builder form. Let us discuss each of these terms in detail.

1. | Sets Definition |

2. | Sets Representation |

3. | Sets Symbols |

4. | Types of Sets |

5. | Sets Formulas |

7. | Sets Properties |

8. | Operations on Sets |

9. | FAQs on Sets |

## Sets Definition

In mathematics, a set is a well-defined collection of objects. Sets are named and represented using a capital letter. In the set theory, the elements that a set comprises can be any kind of thing: people, letters of the alphabet, numbers, shapes, variables, etc.

### Sets in Maths Examples

We know that a collection of even natural numbers less than 10 is defined, whereas collection of intelligent students in a class is not defined. Thus, collection of even natural numbers less than 10 can be represented in the form of a set, A = {2, 4, 6, 8}. Let us use this example to understand the basic terminology associated with sets in math.

### Elements of a Set

The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas. To denote that an element is contained in a set, the symbol '∈' is used. In the above example, 2 ∈ A. If an element is not a member of a set, then it is denoted using the symbol '∉'. Here, 3 ∉ A.

### Cardinal Number of a Set

The cardinal number, cardinality, or order of a set denotes the total number of elements in the set. For natural even numbers less than 10, n(A) = 4. Sets are defined as a collection of unique elements. One important condition to define a set is that all the elements of a set should be related to each other and share a common property. For example, if we define a set with the elements as the names of months in a year, then we can say that all the elements of the set are the months of the year.

## Representation of Sets

There are different set notations used for the representation of sets. They differ in the way in which the elements are listed. The three set notations used for representing sets are:

- Semantic form
- Roster form
- Set builder form

### Semantic Form

The semantic notation describes a statement to show what are the elements of a set. For example, Set A is the list of the first five odd numbers.

### Roster Form

The most common form used to represent sets is the roster notation in which the elements of the sets are enclosed in curly brackets separated by commas. For example, Set B = {2,4,6,8,10}, which is the collection of the first five even numbers. In a roster form, the order of the elements of the set does not matter, for example, the set of the first five even numbers can also be defined as {2,6,8,10,4}. Also, if there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element. For example, infinite sets are represented as, X = {1, 2, 3, 4, 5 ...}, where X is the set of natural numbers. To sum up the notation of the roster form, please take a look at the examples below.

Finite Roster Notation of Sets : Set A = {1, 2, 3, 4, 5} (The first five natural numbers)

Infinite Roster Notation of Sets : Set B = {5, 10, 15, 20 ....} (The multiples of 5)

### Set Builder Form

The set builder notation has a certain rule or a statement that specifically describes the common feature of all the elements of a set. The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set. For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20. Sometimes a ":" is used in the place of the "|".

### Visual Representation of Sets Using Venn Diagram

Venn Diagram is a pictorial representation of sets, with each set represented as a circle. The elements of a set are present inside the circles. Sometimes a rectangle encloses the circles, which represents the universal set. The Venn diagram represents how the given sets are related to each other.

## Sets Symbols

Set symbols are used to define the elements of a given set. The following table shows some of these symbols and their meaning.

Symbols | Meaning |
---|---|

U | Universal set |

n(X) | Cardinal number of set X |

b ∈ A | 'b' is an element of set A |

a ∉ B | 'a' is not an element of set B |

{} | Denotes a set |

∅ | Null or empty set |

A U B | Set A union set B |

A ∩ B | Set A intersection set B |

A ⊆ B | Set A is a subset of set B |

B ⊇ A | Set B is the superset of set A |

## Types of Sets

Sets are classified into different types. Some of these are singleton, finite, infinite, empty, etc.

### Singleton Sets

A set that has only one element is called a singleton set or also called a unit set. Example, Set A = { k | k is an integer between 3 and 5} which is A = {4}.

### Finite Sets

As the name implies, a set with a finite or countable number of elements is called a finite set. Example, Set B = {k | k is a prime number less than 20}, which is B = {2,3,5,7,11,13,17,19}

### Infinite Sets

A set with an infinite number of elements is called an infinite set. Example: Set C = {Multiples of 3}.

### Empty or Null Sets

A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅'. It is read as '**phi**'. Example: Set X = {}.

### Equal Sets

If two sets have the same elements in them, then they are called equal sets. Example: A = {1,2,3} and B = {1,2,3}. Here, set A and set B are equal sets. This can be represented as A = B.

### Unequal Sets

If two sets have at least one element that is different, then they are unequal sets.Example: A = {1,2,3} and B = {2,3,4}. Here, set A and set B are unequal sets. This can be represented as A ≠ B.

### Equivalent Sets

Two sets are said to be equivalent sets when they have the same number of elements, though the elements are different. Example: A = {1,2,3,4} and B = {a,b,c,d}. Here, set A and set B are equivalent sets since n(A) = n(B)

### Overlapping Sets

Two sets are said to be overlapping if at least one element from set A is present in set B. Example: A = {2,4,6} B = {4,8,10}. Here, element 4 is present in set A as well as in set B. Therefore, A and B are overlapping sets.

### Disjoint Sets

Two sets are disjoint sets if there are no common elements in both sets. Example: A = {1,2,3,4} B = {5,6,7,8}. Here, set A and set B are disjoint sets.

### Subset and Superset

For two sets A and B, if every element in set A is present in set B, then set A is a subset of set B(A ⊆ B) and B is the superset of set A(B ⊇ A).

Example: A = {1,2,3} B = {1,2,3,4,5,6}

A **⊆** B, since all the elements in set A are present in set B.

B ⊇ A denotes that set B is the superset of set A.

### Universal Set

A universal set is the collection of all the elements in regard to a particular subject. The universal set is denoted by the letter 'U'. Example: Let U = {The list of all road transport vehicles}. Here, a set of cars is a subset for this universal set, the set of cycles, trains are all subsets of this universal set.

### Power Sets

Power set is the set of all subsets that a set could contain. Example: Set A = {1,2,3}. Power set of A is = {{∅}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}.

## Sets Formulas

Sets find their application in the field of algebra, statistics, and probability. There are some important set formulas as listed below.

For any two overlapping sets A and B,

- n(A U B) = n(A) + n(B) - n(A ∩ B)
- n (A ∩ B) = n(A) + n(B) - n(A U B)
- n(A) = n(A U B) + n(A ∩ B) - n(B)
- n(B) = n(A U B) + n(A ∩ B) - n(A)
- n(A - B) = n(A U B) - n(B)
- n(A - B) = n(A) - n(A ∩ B)

For any two sets A and B that are disjoint,

- n(A U B) = n(A) + n(B)
- A ∩ B = ∅
- n(A - B) = n(A)

## Properties of Sets

Similar to numbers, sets also have properties like associative property, commutative property, and so on. There are six important properties of sets. Given, three sets A, B, and C, the properties for these sets are as follows.

Property | Example |
---|---|

Commutative Property | A U B = B U A A ∩ B = B ∩ A |

Associative Property | (A ∩ B) ∩ C = A ∩ (B ∩ C) (A U B) U C = A U (B U C) |

Distributive Property | A U (B ∩ C) = (A U B) ∩ (A U C) A ∩ (B U C) = (A ∩ B) U (A ∩ C) |

Identity Property | A U ∅ = A A ∩ U = A |

Complement Property | A U A' = U |

Idempotent Property | A ∩ A = A A U A = A |

## Operations on Sets

Some important operations on sets include union, intersection, difference, the complement of a set, and the cartesian product of a set. A brief explanation of operations on sets is as follows.

### Union of Sets

Union of sets, which is denoted as A U B, lists the elements in set A and set B or the elements in both set A and set B. For example, {1, 3} ∪ {1, 4} = {1, 3, 4}

### Intersection of Sets

The intersection of sets which is denoted by A ∩ B lists the elements that are common to both set A and set B. For example, {1, 2} ∩ {2, 4} = {2}

### Set Difference

Set difference** **which is denoted by A - B, lists the elements in set A that are not present in set B. For example, A = {2, 3, 4} and B = {4, 5, 6}. A - B = {2, 3}.

### Set Complement

Set complement which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.

### Cartesian Product of Sets

The cartesian product** **of two sets which is denoted by A × B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained. For example, {1, 3} × {1, 3} = {(1, 1), (1, 3), (3, 1), (3, 3)}.

**☛ ****Topics Related to Sets:**

Check out some interesting topics related to sets.

- Operations on Sets
- Venn Diagrams
- Subset
- Roster Notation
- Universal Set
- Intersection of Sets
- Set Builder Notation

## FAQs on Sets

### What is Sets in Mathematics and Examples?

Sets are a collection of distinct elements, which are enclosed in curly brackets, separated by commas. The list of items in a set is called the elements of a set. Examples are a collection of fruits, a collection of pictures. In another way, sets are represented as follows. Set A = {a,b,c,d}. Here, a,b,c,d are the elements of set A.

### What are Different Sets Notations to Represent Sets?

Sets can be represented in three ways. Representing sets means a way of listing the elements of the set. They are as follows.

- Semantic Notation: The elements of a set are represented by a single statement. For example, Set A is the number of days in a week.
- Roster Notation: This form of representation of sets uses curly brackets to list the elements of the set. For example, Set A = {2,4,6,8,10}
- Set Builder Notation: A set builder form represents the elements of a set by a common rule or a property. For example, {x | x is a prime number less than 20}

### What are the Types of Sets?

Sets differ from each depending upon elements present in them. Based on this, we have the following types of sets. They are singleton sets, finite and infinite sets, empty or null sets, equal sets, unequal sets, equivalent sets, overlapping sets, disjoint sets, subsets, supersets, power sets, and universal sets.

### What are the Properties of Sets in Set-Theory?

Different properties associated with sets in math are,

- Commutative Property: A U B = B U A and A ∩ B = B ∩ A
- Associative Property: (A ∩ B) ∩ C = A ∩ (B ∩ C) and (A U B) U C = A U (B U C)
- Distributive Property: A U (B ∩ C) = (A U B) ∩ (A U C) and A ∩ (B U C) = (A ∩ B) U (A ∩ C)
- Identity Property: A U ∅ = A and A ∩ U = A
- Complement Property: A U A' = U
- Idempotent Property: A ∩ A = A and A U A = A

### What is the Union of Sets?

The union of two sets A and B are the elements from both set A and B, or both combined together. It is denoted using the symbol 'U'. For example, if set A = {1,2,3} and set B = {4,5,6}, then A U B = {1,2,3,4,5,6}. A U B is read as 'A union B'.

### What is the Intersection of Sets?

The intersection of two sets A and B are the elements that are common to both set A and B. It is denoted using the symbol '∩'. For example, if set A = {1,2,3} and set B = {3,4,5}, then A ∩ B = {3}. A ∩ B is read as 'A intersection B'.

### What are Subsets and Supersets?

If every element in a set A is present in set B, then set B is the superset of set A and set A is a subset of set B.

Example: A = {1,4,5} B = {1,2,3,4,5,6}

Since all elements of set A are present in set B. ⇒ A ⊆ B and B ⊇ A.

### What are Universal Sets?

A universal set, denoted by the letter 'U', is the collection of all the elements in regard to a particular subject.

Example: Let U = {The list of all road transport vehicles}. Here, a set of cycles is a subset of this universal set.

### What is Complement in Sets?

The complement of a set which is denoted by A', is the set of all elements in the universal set that are not present in set A. In other words, A' is denoted as U - A, which is the difference in the elements of the universal set and set A.

### What is Cartesian Product in Sets?

Cartesian product of two sets, denoted by A×B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained. For example, if A = {1,2} and B = {3,4}, then A×B = {(1,3), (1,4), (2,3), (2,4)}.

### What is the Use of Venn Diagram in Sets?

Venn Diagram is a pictorial representation of the relationship between two or more sets. Circles are used to represent sets. Each circle represents a set. A rectangle that encloses the circles represents the universal set.

## FAQs

### What is set and types of set with example? ›

...

Related Links.

Disjoint Set | Power Set |
---|---|

Complement Of Set | Subsets |

Types Of Sets | Union And Intersection Of Sets Cardinal Number Practical Problems |

**What is set types of sets and their symbols? ›**

Symbol | Symbol Name | Meaning |
---|---|---|

A ∪ B | union | Elements that belong to set A or set B |

A ∩ B | intersection | Elements that belong to both the sets, A and B |

A ⊆ B | subset | subset has few or all elements equal to the set |

A ⊄ B | not subset | left set is not a subset of right set |

**What are the 12 types of sets? ›**

**What is Set, What are Types of Sets, and Their Symbols?**

- Empty Sets. The set, which has no elements, is also called a null set or void set. ...
- Singleton Sets. The set which has just one element is named a singleton set. ...
- Finite and Infinite Sets. ...
- Equal Sets. ...
- Subsets. ...
- Power Sets. ...
- Universal Sets. ...
- Disjoint Sets.

**What are the properties of set with example? ›**

In Mathematics, a set is defined as **a collection of well-defined objects**. For example, the set of natural numbers between 1 and 10, the set of even numbers less than 20. If we change the order of writing the elements in a set, it does not make any changes in the set.

**What are examples of sets? ›**

...

**Some commonly used sets are as follows:**

- N: Set of all natural numbers.
- Z: Set of all integers.
- Q: Set of all rational numbers.
- R: Set of all real numbers.
- Z
^{+}: Set of all positive integers.

**What are all the types of sets? ›**

The different types of sets are empty set, finite set, singleton set, equivalent set, subset, power set, universal set, superset and infinite set. Let us learn about the types of sets with examples.

**What are the 4 sets of numbers? ›**

Whole Numbers - The set of Natural Numbers with the number 0 adjoined. Integers - Whole Numbers with their opposites (negative numbers) adjoined. Rational Numbers - All numbers which can be written as fractions. Irrational Numbers - All numbers which cannot be written as fractions.

**How do you solve a set in math? ›**

What Is the Formula of Sets? The set formula is given in general as **n(A∪B) = n(A) + n(B) - n(A⋂B)**, where A and B are two sets and n(A∪B) shows the number of elements present in either A or B and n(A⋂B) shows the number of elements present in both A and B.

**What do the set symbols mean? ›**

set. The symbol **∈ means “is an element of”** **The symbol /∈ means “is not an element of”** Generally capital letters are used to represent sets and lowercase. letters are used for other objects i.e. S = {2, 3, 5, 7}

**What is set math grade 7? ›**

A set is **a collection of unique objects** i.e. no two objects can be the same. Objects that belong in a set are called members or elements.

### What are the basic sets in math? ›

**Finite set: The number of elements is finite**. **Infinite set: The number of elements are infinite**. **Empty set: It has no elements**. **Singleton set: It has one only element**.

**What are the 5 sets of numbers? ›**

**Types of numbers**

- Natural Numbers (N), (also called positive integers, counting numbers, or natural numbers); They are the numbers {1, 2, 3, 4, 5, …}
- Whole Numbers (W). ...
- Integers (Z). ...
- Rational numbers (Q). ...
- Real numbers (R), (also called measuring numbers or measurement numbers).

**What are the 5 properties of math and examples? ›**

**Properties of Mathematics**

- Commutative Property of Addition. When two numbers are added, the sum is the same regardless of the order of the addends. ...
- Commutative Property of Multiplication. ...
- Associative Property of Addition. ...
- Associative Property of Multiplication.

**What are types of properties *? ›**

**Kinds of Property**

- Movable Property. Movable property can be moved from one place to another without causing any damage. ...
- Immovable Property. Immovable property is one that cannot be moved from one place to another place. ...
- Tangible Property. ...
- Intangible Property. ...
- Public Property. ...
- Private Property. ...
- Personal Property. ...
- Real Property.

**What is set Give 5 examples? ›**

Expert-Verified Answer

**The collection of first five natural numbers**. The collection of vowels of the English Alphabet. The collection of whole numbers between 20 and 25. The collection of natural numbers between 30 and 35.

**How do you use a set example? ›**

If you set an example, you **encourage or inspire people by your behaviour to behave or act in a similar way**. An officer's job was to set an example. He is setting an example which other aristocrats and leading Britons should follow.

**What is a set in math definition? ›**

set, in mathematics and logic, **any collection of objects (elements), which may be mathematical (e.g., numbers and functions) or not**. A set is commonly represented as a list of all its members enclosed in braces. The intuitive idea of a set is probably even older than that of number.

**What are sets of 10 called? ›**

A collection of ten items (most often ten years) is called **a decade**. The ordinal adjective is decimal; the distributive adjective is denary.

**What are the 4 operations of sets? ›**

**The four important basic operations of sets are :**

- Union of sets.
- Intersection of sets.
- Complement of sets.
- Cartesian product of sets.

**What are the common sets? ›**

Name | Symbol | Elements of Number Set |
---|---|---|

Composite Numbers | P′ | {4,6,8,9,12,14,15,16,18,20,21,22,23,24,25,26,28,30,32,33,34,35,36,…} |

Whole Numbers | W | {0,1,2,3,4,5,…} |

Integer Numbers | Z | {0,±1,±2,±3,±4,±5,…} |

Rational Numbers | Q | {x∣x=pq,q≠0,p∈Z,q∈Z} |

### What is the 8 set of digits called? ›

The **octal numeral system**, or oct for short, is the base-8 number system, and uses the digits 0 to 7. This is to say that 10_{octal} represents eight and 100_{octal} represents sixty-four.

**What is the rule of 4 in math? ›**

The Rule of Four **stipulates that topics in mathematics should be presented in four ways: geometrically, numerically, analytically, and verbally**. Implementing the Rule of Four supports students in being adept with all four types of representations and also provides support to students who learn in different ways.

**What is the set A ∩ B? ›**

The intersection operation is denoted by the symbol ∩. The set A ∩ B—read “A intersection B” or “the intersection of A and B”—is defined as **the set composed of all elements that belong to both A and B**. Thus, the intersection of the two committees in the foregoing example is the set consisting of Blanshard and Hixon.

**What is A∩B example? ›**

For any two sets A and B, the intersection, A ∩ B (read as A intersection B) lists all the elements that are present in both sets, and are the common elements of A and B. For example, **if Set A = {1,2,3,4,5} and Set B = {3,4,6,8}, A ∩ B = {3,4}**.

**What is the rule of set? ›**

Both the universal set and the empty set are subsets of every set. Rule is **a method of naming a set by describing its elements**. { x: x > 3, x is a whole number} describes the set with elements 4, 5, 6,…. Therefore, { x: x > 3, x is a whole number} is the same as {4,5,6,…}.

**What is symbol in math? ›**

Symbols are images that represent something. In general, symbols are determined by their context. In mathematics, symbols generally **represent operations or relationships between numbers or values**.

**How do you read a math set? ›**

In Mathematics, the set is **an unordered group of elements represented by the sequence of elements (separated by commas) between curly braces {" and "}**. For example, {cat, cow, dog} is a set of domestic animals, {1, 3, 5, 7, 9} is a set of odd numbers, {a, b, c, d, e} is a set of alphabets.

**What is the set Z *? ›**

Integers. **The set of integers** is represented by the letter Z.

**What is Z set in math? ›**

Special sets

Z denotes the set of integers; i.e. {…,−2,−1,0,1,2,…}. Q denotes the set of rational numbers (the set of all possible fractions, including the integers). R denotes the set of real numbers.

**How many elements in a set? ›**

A set may have **infinitely many elements**, so we can't list all of them. For example let E = {all even integers greater than or equal to 1}. We write this as E = {2,4,6,...}, where “...” should be read as “et cetera”.

### What are the 6 types of numbers? ›

**Main types**

- Natural numbers ( ...
- Integers ( ...
- Rational numbers ( ...
- Real numbers ( ...
- Irrational numbers: Real numbers that are not rational.
- Imaginary numbers: Numbers that equal the product of a real number and the square root of −1. ...
- Complex numbers (

**How many is a set of numbers? ›**

A set of numbers is a collection of numbers, called elements. The set can be **either a finite collection or an infinite collection of numbers**.

**What is set explain? ›**

set, in mathematics and logic, any collection of objects (elements), which may be mathematical (e.g., numbers and functions) or not. A set is commonly represented as a list of all its members enclosed in braces. The intuitive idea of a set is probably even older than that of number.

**What is set in simple word? ›**

A set is **a group of things that belong together**, like the set of even numbers (2,4,6…) or the bed, nightstands, and dresser that make up your bedroom set. Set has many different meanings. As a verb, it means to put in place.

**What is set definition in mathematics? ›**

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

**What is set equation? ›**

Formulas of Sets

If there are two sets P and Q, n(P U Q) represents the number of elements present in one of the sets P or Q. n(P ⋂ Q) represents the number of elements present in both the sets P & Q. **n(P U Q) = n(P) + (n(Q) – n (P ⋂ Q)**

**How do you use set? ›**

**20 different ways to use the word SET in English**

- set = physically put something in a position. ...
- set = put something into a certain state. ...
- set = adjust controls on a device. ...
- set = decide on a date/price. ...
- set = establish. ...
- set (adjective) = fixed or established. ...
- set (adjective) = ready. ...
- set (noun) = a group of similar things.

**What is the set answer? ›**

Answer: A set is **a group or collection of objects or numbers, considered as an entity unto itself**.

**What are the parts of a set? ›**

**The objects in a set are called the elements (or members ) of the set**; the elements are said to belong to the set (or to be in the set), and the set is said to contain the elements. Usually the elements of a set are other mathematical objects, such as numbers, variables, or geometric points.

**What is a set in algebra? ›**

A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. Every item in the set is called an element of the set. Curly brackets are used while writing a set. A very simple example of a set would be like this. Set A = {1,2,3,4,5}.

### What type of word is set? ›

**verb (used with object)**, set, set·ting. to put (something or someone) in a particular place: to set a vase on a table.

**What are the properties of sets? ›**

The six properties of sets are commutative property, associative property, distributive property, identity property, complement property, idempotent property.

**What is empty set with example? ›**

**A set which does not contain any element** is called the empty set or the null set or the void set. For example, the set of the number of outcomes for getting a number greater than 6 when rolling a die. As we know, the outcomes of rolling a die are 1, 2, 3, 4, 5, and 6.