Set Theory Formulas

In mathematics, a set is simply a collection of well-defined individual objects that form a group. A set can contain any group of items, such as a set of numbers, a day of the week, or a vehicle. Each element of the set is called an element of the set. Curly braces are used to create sets. A very simple example of a set is: Set A = {1,2,3,4,5}. There are various notations for representing the elements of a set. Sets are typically expressed using either a list form or a set builder form.

In mathematics, a set is defined as a collection of immutable objects with fixed elements. Elements cannot be repeated in a set but can be written in any order. Sets are indicated by capital letters. In set theory, the elements that make up a set can be anything: a person, an alphabetic letter, a number, a shape, or a variable.

We know that the set of even natural numbers less than 20 is defined, whereas the set of smart students of the class is undefined. So, the set of even natural numbers less than 20 can be written as the set A = {2, 4, 6, 8, 10, 12, 14, 16, 18 }.

Elements Of Sets

Let’s take an example.   A = { 2, 4, 6, 8 } . A is a set and  2, 4, 6, and 8 are elements of the set or members of the set. Elements written in a set can be used in any order, but cannot be repeated. All elements of a set are represented by lowercase letters in the alphabet. 

We can also write this as 2 ∈ A, 4 ∈ A, etc. The cardinality of the set is 4. Here are some commonly used sets. 

1. N: Set of all-natural numbers
2. Z: Set of all integers
3. Q: Set of all rational numbers
4. R: Set of all real numbers
5. Z⁺: Set of all positive integers.

Cardinal Number Or Cardinality Of A Set

The Cardinal number, cardinality, or order of a set indicates the total number of elements in the set. For natural even numbers less than 10, n(A) = 4. A set is defined as a unique collection of elements. One of the important conditions for defining a set is that all elements of the set must have common properties. 

eg. A = { 2, 4, 6, 8 } , cardinality of set A is 4.

Representation Of A Set

There are various set notations used to represent sets, the way the elements are enumerated is different. Three set notations are used to represent sets. 

• 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 even numbers.                                                                                                                                                                                                                                                          
• Roster Form: The most common form used to represent a set is roster form, in which the elements of the set are enclosed in comma-separated braces. Example :   For finite Set: Set A = { 0, 1, 2, 3, 4 } (The first five whole numbers)
   For infinite Set : Set B = { 2, 4, 6, 8 ….  } ( multiples of 2)                                                                                                                                                                                            
• Set builder form: A method of defining a set by describing its properties rather than listing its elements is known as set builder notation.  Set building in  Set Builder notation is also known as set understanding, set abstraction, and set intentions. Set builder notation contains one or more variables and rules for determining which elements belong to and which do not belong to a set. This rule is often expressed in the form of a predicate. Set rules and variables are separated by a vertical slash “|”. or a colon (:). This method is widely used to describe infinite sets. For example, A = { k | k is an odd number, k ≤ 20}

Visual Representation of Sets Using Venn Diagram: A Venn diagram is a graphical representation of various sets. This helps students visualize logical relationships between different sets. Venn diagrams make it easy to understand the differences and similarities between sets. A Venn diagram represents the elements of a set, usually represented by circles and the overlapping circles represent the common elements of the set.                                                                             

Above figure is the visual representation of sets A and B given by A = { a, b, c } and B = { c, d, e } .

Types Of Sets

Singleton Sets

A set with only one element is called a singleton set.

Example: A = { m | m is a natural number between 3 and 5} which is A = {4}.

Empty Sets

A set having no elements is called an empty or null set. An empty set is represented by the “∅” symbol. It reads like “phi”.

Example: B = {x : 1 < x < 2, x is an integer}.

Finite Sets And Infinite Sets

A set consisting of a finite number of elements is said to be a finite set, whereas a set whose elements cannot be evaluated is said to be an infinite set. 

Example: set A = { 1, 5, 9 } is a finite set, as it has a finite number of elements.
Set C (set of natural numbers) = { 1, 2, 3, 4, 5, ……….} is an infinite set.

Equivalent Sets

If two different sets have the same number of elements, they are said to be equivalent sets. 

Example: If A = { 3, 5 , 7 , 9 } and B = { a, b, c, d } 

 Both sets A and B have 4 elements. So set A and set B are equivalent set.

Equal Sets

Two sets A and B are said to be equal if they consist of the same elements and the order of the elements does not matter. 

Example: let X = { 2, 4, 6, 8 } and Y = { 6, 2, 4, 8},  then X = Y .

Unequal Sets

Two sets are unequal if they have at least one other element.

Example: let X = { 2, 5 , 6, 8 } and Y = { 6, 2, 4, 8},  then  X  and Y  are unequal sets .

Disjoint  Set

Two sets A and B are said to be disjoint if they do not contain a single common element.

Example: let X= { 5, 6, 7, 8 } and Y = {3, 9, 12, 15 }. Here, set X and set Y are disjoint sets.

Subset and Superset

For two sets A and B, if all elements of set A are in set B, then set A is a subset of set B (A ⊆ B) and B is a superset of set A (B ⊇ A).

Example: A= { 2, 4, 6 }

So, { 2, 4 } ⊆ A.

Other subsets of set A are: { 2 }, { 4 }, { 6 } , { 2,6 }, { 4, 6 } ,{ 2,4,6 },{ }.

Proper Subset

For two sets A and B, if A is a subset of  B and A is not equal to B, then A is a proper subset of B.

Example: A= { 2, 4, 6 } and B = { 2, 7, 5, 4, 6 }, here A is a subset of set B and A is not equal to B, so A is a proper subset of B.

Universal Set

A universal set is a collection of all elements related to a particular subject. Universal sets are denoted by the letter “U”.

Example: If X = { 2, 4 } and U = { 2, 3, 4, 5 } , Then U = {1,2,3,4,5} is the universal set .

Power Set

A set containing all the subsets of a set is called the power set of that set.

Example: A= { 2, 4, 6 } then P(A) = { { 2 }, { 4 }, { 6 }, { 2,4 }, { 2,6 }, { 4, 6 }, { 2,4,6 }, { } } .

Operation On Sets 

Union of Sets

The union of sets denoted by A U B lists the elements of sets A or  B at least one time.

Example: If A= { 2, 4, 6 } and B = { 2, 7, 5, 4, 6 } then , A U B = { 2, 4, 5, 6, 7 }.

Intersection of Sets

The intersection of the sets denoted by A ∩ B contains a list of elements common to sets A and  B.

Example: If A= { 2, 4, 6 } and B = { 2, 7, 5, 4, 6 } , then A ∩ B = { 2, 4, 6 } .

Difference of sets

The set difference, denoted A – B, lists the elements of set A that are not in set B.

Example: If  A = { 2, 7, 5, 4, 6 }  and B = { 2, 4, 6 } , then A – B = { 5, 7 } .

Complement of a Set

The complement of the set denoted by A’ is the set of all elements of the universal set that do not exist in set A. That is, A’ is denoted as U – A, which is the difference between elements of the universal set and set A.

Cartesian Product of Sets

The Cartesian product of two sets, denoted  A × B, is the product of two nonempty sets that yields an ordered pair of elements.

Example: If A= { 1, 3 } and B = { 2, 4 } then , A × B = { (1,2) , (1,4) , (3,2) , (3,4) } .

Set Formulas

A set formula is a formula related to set theory in mathematics. A set is a well-defined collection of objects made up of individual elements. Set knowledge helps you apply set formulas in areas related to statistics, probability, geometry, and sequences.

Set Formulas On Number Of elements of Sets

If n(A) and n(B) represent the number of elements in two finite sets A and B, respectively, then

• 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)

If sets A and B are disjoint sets then

• n(A U B) = n(A) + n(B)
• A ∩ B = ∅
• n(A – B) = n(A)

Sets Formulas on Properties of Sets 

Set formulas have almost the same properties as real or natural numbers. Aggregations also have commutative, associative, and distributive properties. The set formula according to the properties of the set is as follows.

• Commutativity

A⋂ B = B⋂ A 
A∪ B = B∪ A

• Associativity

A⋂ (B⋂ C) = (A⋂ B)⋂ C
A∪ (B∪ C) = (A∪ B)∪ C

• Distributivity

A ⋂ (B∪ C) = (A ⋂ B) ∪ (A⋂ C)

• Idempotent Law

A ⋂ A = A
A ∪ A = A

• Law of Ø and U

A⋂ Ø = Ø
U ⋂ A = A
A ∪ Ø = A
U ∪ A = U

Sets Formulas of Complement Sets

The set formulas of set complement include the basic law of complement, De Morgan’s law, double complement, and the law of empty and universal sets.     

1. Complement Law                                      A∪A’ = U, A⋂A’ = Ø and A’ = U –  A
2. De Morgan’s Laws                                    (A ∪B)’ = A’ ⋂B’ and (A⋂B)’ = A’ ∪ B’
3. Law of Double complementation            (A’)’ = A
4. Laws of Empty set and Universal Set       Ø’ = ∪ and ∪’ = Ø

Sets Formulas of Difference of Sets

1. A – A = Ø
2. B – A = B⋂ A’
3. B – A = B – (A⋂B)
4. (A – B) = A if A⋂B =  Ø
5. (A – B) ⋂ C = (A⋂ C) – (B⋂C)
6. A ΔB = (A-B) U (B- A) 
7. n(AUB) = n(A – B) + n(B – A) + n(A⋂B)
8. n(A – B) =  n(A∪B) – n(B)
9. n(A – B) = n(A) – n(A⋂B)
10. n(A’) = n(∪) – n(A)

Sample Questions

Question 1: What is a set in mathematics? Give examples.

Answer:

A set is a collection of individual elements enclosed in braces and separated by commas.  Example: Collecting vegetables, collecting notebooks. Alternatively, the set can be expressed as set A = { 1, 2, 3, 4 } where 1, 2, 3, and 4 are elements of set A.

Question 2: Why do we use sets in math? 

Answer: 

The purpose of using sets is to represent a set of related objects in a group. In mathematics, we usually refer to groups of numbers, such as groups of natural numbers, sets of rational numbers, etc.

Question 3: What is the union of sets?  

Answer:

The union of two sets A and B contains elements of both sets A and B . It is denoted by the symbol “U”. For example, if  set A = { 2,3 } and  B = { 5,6 } then AUB = { 2,3,5,6 }. 

Question 4: How can we represent the given set in set-builder form A = {1, 3, 5, 7, 9}

Answer:

We can represent the given set in set-builder form as A = { x | x is an odd natural number less than 10 }.

Question 5: Given A = { 10, 12, 14, 16, 18 } and B = { 14, 16 }. Find A U B and A ⋂ B and A – B.

Answer:

As , A = { 10, 12, 14, 16, 18 } and B = { 14, 16 }

A U B = { 10, 12, 14, 16, 18 }

A ⋂ B = { 14, 16 }

A – B = { 10, 12, 18 }  

Question 6: What is the formula for the intersection of sets?

Answer:

The set expression for the intersection of sets A and B is denoted by ⋂ and n(A⋂B) denotes the elements common to sets A and B. So,  formula for intersection of sets is given by  n(A⋂B) = n(A) + n(B)- n(A∪B) .

Question 7: What is the application of the set formulas?  

Answer:

Set formulas have broad application in many abstract concepts. 

For example, if R is the set of real numbers and Q is the set of rational numbers, then R – Q is the set of irrational numbers. 

Probability theory adopts set rules.  For example, a sample space is a universal set. If A and B are two mutually exclusive events, then P(A∪B) = P(A) + P(B) – P(A⋂B).

Question 8: There are 240 students in the class, 92 playing badminton, 60  table tennis, 80 rugby, 27 playing badminton and table tennis, 20 playing rugby and table tennis, 16 playing badminton and rugby, and 60 playing all the 3 games.  Find  

1. Number of students who play badminton, table tennis, and rugby.
2. Number of students who play badminton and not rugby.
3. Number of students who play badminton and rugby and not table tennis.

Answer:

Consider, that n(U) is the total number of students in class and n(B), n(T), and n(R) is the number of students who play badminton, table tennis, and rugby respectively.

Here, n(B∩T) is the number of students who play both badminton and tennis, n(R∩T) is the number of students who play both rugby and tennis and n(B∩R) is the number of students who play both badminton and rugby. 

We are given that: n(U) = 240 , n(B)=92 , n(T)=60 , n(R)=80 ,  n(B∩T)=27 , n(R∩T)=20, n(B∩R)=16 and n(B’∩T’∩R’) ( number of students which do not play any of the three games ) = 60   

Given  n(B’∩T’∩R’)=60  ⇒n(B∪T∪R)’=60

So,  n(B∪T∪R) = n(U) -n(B∪T∪R)’ = 240-60=180

Now , n(B∪T∪R) =  n(B) + n(T) + n(R) – n(B∩T) – n(R∩T) – n(B∩R) + n(B∩T∩R)

180= 92+60+80-27-20-16+n(B∩T∩R)

n(B∩T∩R) =180+63-232=243-232=11

1. Number of students who play badminton, table tennis and rugby = 11.
2. Number of students who play badminton but not rugby, n(B-R)  =  n(B)-n(B∩R)=92-16=76
3. Number of students who play badminton and rugby but not table tennis ,n(B∩R∩T’)= n(B∩R)-n(B∩R∩T) =16-11=5.

Question 9: In a survey of 800 students in a school, 250 students were found to be drinking mojito and 500 were drinking juice, 150 were drinking both mojito and juice. Find how many students were drinking neither mojito nor juice.

Answer:

Consider n(U) is the total number of students in school , n (M) is the number of students which are drinking mojito , n (J) is the number of students which are drinking juice , n(M∩J) are the number of students which are drinking both mojito and juice

Given  , n(U) = 800, n(M)=250,n(J)=500,n(M∩J) =150

We have to find number of students which are drinking neither mojito nor juice , n(M’∩J’) =  n(M∪J)’

n(M∪J)’ =n(u)-n(M∪J) 

n(M∪J) =n(M) +n(J)-n(M∩J)=250+500-150=600

 n(M∪J)’ = 800-600=200.