What is a set?

In daily life, we use words like a dinner set, a set of players, a set of books, etc. What is meant by the word ‘set’ here? It clearly defines a collection of similar objects or things. In math, sets are also defined as similar contexts.

## Set Definition

Sets in Mathematics are defined as a collection of well-organized objects or elements of a real or imaginary world. For example, the set of two books: Mathematics and English, the set of the first 10 even natural numbers, a collection of novels, types of animals, and so on. Sets are usually denoted by the capital letters of the English alphabet like A, B, C, … …, X, Y, Z.

For example, the set of five numbers 2, 3, 4, 5, 7 can be written as A = {2, 3, 4, 5, 7}

What is a ‘well organized’ object?

Consider an example. A collection of natural numbers less than 10 is defined, but a collection of brave students in a class is not.

Elements of a Set

Each object or member of the set is called its element. They are enclosed in curly brackets and separated by commas. The elements that are written in the set can be in any order but cannot be repeated. All the set elements are represented in small letters in the case of alphabets. An element is contained in a set, we use the symbol “∈”. It is read as ‘belongs to’. Also, if an element is not contained in a set, we use the symbol “∉”. It is read as ‘does not belong to’.

Suppose we have a set of prime numbers less than 10.

X = {2, 3, 5, 7}.

Here, 2 ∈ A and we read as 2, is a member of X (2 belongs to X)

The set X does not contain 9. So it is expressed as 9 ∉ X and is read as 9 is not a member of X (9 does not belong to X)

The number of elements in the set is denoted by n(B) where B is a set.

Example: B = {2, 3, 4, 5, 7, 8, 9}.

n(B) = 7.

The other word used for the number of elements in the set is also known as its cardinality.

## Representation of Sets

Sets can represent two forms, namely Roster Form or Tabular Form and Set Builder Form. The two forms only difference is in the way in which the elements are listed.

#### Roster Form

The most familiar and easy way to represent sets is the roster form. In this form, all elements or members are listed and put inside curly braces,{ } and if there is more than one element, then elements are separated by commas. For example A = {1, 2}, B = {a, b, c}, N = {1, 2, 3, 4, 5, … ,100} etc. The order of the elements does not matter and a set can contain an infinite number of elements, which we define using a series of dots at the end of the last element.

#### Set Builder Form

In this form, all elements or members are not listed rather their common property is described. The general form of a set is, A = { x: property }

For example, A = {x: x is a prime number, x < 20}.

Here, ‘:’ represent ‘such that’ and use ‘:’ place of the “|” sometimes.

Moreover, In this form element of the set is specified by rules, that the reason the form is also called Rule Method.

## Types of Sets

There are several types of sets in Mathematics. They are empty sets, finite and infinite sets, proper sets, equal sets, etc. Let us go discuss the classification of sets below.

#### Finite set

A set in which elements can be determined by counting is called a finite set.

Example: A set of prime numbers up to 20.

A = {2, 3, 5, 7, 11, 13, 17, 19}

#### Infinite set

Simply, a set that is not finite is known as an infinite set. A set in which elements can not be determined by counting is called an infinite set

Example: A set of all whole numbers.

A = {0, 1,2,3,4,5,6,7,8,9……}

#### Singleton Sets

A set has only one element, it is called a singleton set.

Example: There is only one apple in a basket of grapes.

Another Example: A = {x : x is a natural number and 2 < x < 4} = {3}.

#### Empty Sets

When a set does not have any element is said to be an empty set or void set or null set. It is denoted by the symbol { } or Ø.

Example: A = {y: y is an integer number and 19 < y < 20} = Ø.

#### Equal Sets

Two sets are said to be equal if they have the same elements, the order of elements does not matter.

Example: Two sets A = {1, 4, 5} and B = {1 ,4, 5}. Here, A = B

#### Unequal Sets

Two sets are said to be unequal if they have at least one different element.

Example: Two sets A = {3, 4, 5} and B = {1 ,4, 5}. Here, Here, A ≠ B as 3 ∈ A but 3 ∉ B.

#### Equivalent Sets

If the number of elements is the same for two different sets, even if the elements are different, then they are called equivalent sets.

It is represented as:

n(A) = n(B)

Example: Two sets A = {3, 7 ,8} and B = {Red, Blue, Green}. Since n(A) = n(B), A and B are equivalent sets.

#### Subset and Superset

If A and B are sets then A will be a subset of B if and only if every element of A are also elements of B, denoted as A ⊆ B B is known as the superset of set A, which is denoted by B ⊇ A.

Example: A = {4, 7} and B = {4, 6, 7, 8}

A ⊆ B, since all the elements of A are contained in B.

Also, B ⊇ A, set B is the superset of set A.

#### Proper Subset

A is a proper subset of B if and only if A ⊆ B and A ≠ B. That means every element of A is also an element of B and B has at least one element which is not an element of A. It can be written as A ⊂ B.

Example: A = {a, b} is a subset of B = {a, b, c}. Here A ⊂ B.

#### Universal Set

A set that contains all sets relevant to a particular condition is called a universal set. This is the set of all possible values. It is expressed by the letter “U.”

Example: U is the set of all the natural numbers. So, the set of even numbers, set of odd numbers, and set of prime numbers is a subset of U which means a universal set.

#### Disjoint Sets

Two sets A and B are called disjoint sets if they have no common elements in both sets.

Example: A = {3, 4, 5, 9} and B = {2, 6, 7, 8} are disjoint sets because there is no common element in two sets.

#### Power Sets

The set of all subsets of A is called the power set of A and it is represented by P(A)

Example: Suppose we have a set A = {a, b}. Power set of A is = {{∅}, {a}, {b}, {a,b}}.

## Operations on Sets

The operations of the sets in mathematics are performed when two or more sets are combined to form a single set under some given conditions. The basic set operations are given below:

- Union of Sets
- Intersection of Sets
- Complement of a Set
- Difference of Set
- Cartesian Product of Sets

#### Union of Sets

If A and B are two sets then the union of sets A and B is the set that contains all elements of sets A and B and it is denoted by A ∪ B

Example: Two sets A= {3, 4, 5} and B = {4, 6, 8}

A U B = {3, 4, 5, 6, 8}

#### Intersection of Sets

If A and B are two sets then the intersection of sets A and B is the set that contains only the common elements between sets A and B and it denoted by A ∩ B

Example: Two sets A= {3, 4, 5} and B = {4, 6, 8}

A ∩ B = { 4 }

#### Complement of a Set

Let U be the universal set and A be a subset of U. The complement set A, is the set of all elements in the universal set that are not in the set. The complementary set of A denoted by Ac or A’. Mathematically Ac = U \ A

Example: Suppose, the universal set U = {1, 2, 3, 4, 5, 6, 7, 8} and set A = {2, 4, 5}

Ac = {1, 3, 6, 7, 8}

#### Difference of Sets

If set A and set B are two sets then the difference of sets A and B is denoted by A – B or A \B, which has the elements in set A that are not present in set B.

Example: Two sets A = {1, 3, 4, 7} and B ={ 1, 2, 4, 5, 7}

A – B = {3}

#### Cartesian Product of sets

If set A and set B are two sets then the Cartesian product of sets A and set B is a set of all ordered pairs (a,b), such that a ∈ A and b ∈ B. It is denoted by A × B.

In set building form: A × B = {(a, b) : a ∈ A and b ∈ B}

Example: Let, A = {a, b, c} and B = {1, 2, 3}.

A × B = {(a, 1), (b, 1), (c, 1), (a, 2), (b, 2), (c, 2), (a, 3), (b, 3), (c, 3)}

## Important Sets Formulas

Some of the most important formulas are:

For any sets A and B, n(A U B) = n(A) + n(B) – n(A ∩ B) |

For any sets A and B, n(A − B) = n(A U B) − n(B) |

For any sets A and B, n(A − B) = n(A) − n(A ∩ B) |

## Some Proposition of Sets

Here in every case U is the universal set and sets A, B, and C are the subsets of U.

Set Proposition | Representation |

Commutative Law : | A∪B = B∪A A∩B = B∩A |

Associative Law : | A ∪ ( B ∪ C) = ( A ∪ B) ∪ C A ∩ ( B ∩ C) = ( A ∩ B) ∩ C |

Distributive Law : | A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |

De Morgan’s Law : | ( A ∪ B)’ = A’ ∩ B’ ( A ∩ B )’ = A’ ∪ B’ |

Others Law: | For any finite set A A ∪ ∅ = A A ∩ ∅ = ∅ A ∪ U = U A ∩U = A A ∪ A = A A ∩ A = A |

Solved Examples

1. A = {x : x, is a divisor of 36}, How many elements are there in the set A?

Solution: Here,

36 = 1 × 36

36 = 2 × 18

36 = 3 × 12

36 = 4 × 9

36 = 6 × 6

A = {1, 2, 3, 4, 6, 9, 12, 18, 36}

n(A) = 9

2. Express B = {4, 8, 12, 16, 20} in set builder form.

Solution: Here, each element of set B is a multiple of 4 and not exceeding 20.

B = {x : x, is a multiple of 4 and 0 < x < 21}

3. Write the set C = {x: x is a factor 12} in tabular form.

Solution: Here, factors of 12 are 1, 2, 3, 4, 6, 12

Therefore, C = {1, 2, 3, 4, 6, 12}

4. Find A U B and A ⋂ B and A – B. Where A = {1, 2, 3, 4} and B = {2, 4, 7}

Solution: Given, A = {1, 2, 3, 4} and B = {2, 4, 7}

A U B = {1, 2, 3, 4} U {2, 4, 7} = {1, 2, 3, 4, 7}

A ⋂ B = {1, 2, 3, 4} ⋂ {2, 4, 7} = {2, 4}

A – B = {1, 2, 3, 4} – {2, 4, 7} = {1, 3}

## Frequently Asked Questions(FAQ) on Sets

##### Is {0} an empty set?

No. There is one element inside the brackets. So, it is a singleton set, not an empty set.

##### What are sets in mathematics?

The sets in mathematics are a well-defined collection of distinct objects. These objects can be anything: numbers, people, letters, etc.

##### How do you represent a set?

A set is usually represented using curly braces { } with the elements listed inside, separated by commas. For example, the set of vowels in English is represented as {a, e, i, o, u}.

##### What is the difference between a subset and a proper subset?

A subset is a set where all elements of the first set are also elements of the second set. A proper subset is a subset that does not contain all the elements of the other set, meaning it is smaller in size.

##### How do you define the cardinality of a set?

The cardinality of a set refers to the number of elements in the set. For finite sets, it is a natural number, while for infinite sets, it can be described using concepts like countable and uncountable infinity.

##### What is a Venn diagram and how is it related to sets?

A Venn diagram is a graphical representation of sets, usually depicted as overlapping circles, where each circle represents a set. The areas of overlap represent the intersection of the sets.

##### Can a set have duplicate elements?

No, a set cannot have duplicate elements. Each element in a set is unique, and if an element appears more than once, it is only counted once.