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.
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.
- 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.
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 the 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)
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.
A⋂ B = B⋂ A
A∪ B = B∪ A
A⋂ (B⋂ C) = (A⋂ B)⋂ C
A∪ (B∪ C) = (A∪ B)∪ C
A ⋂ (B∪ C) = (A ⋂ B) ∪ (A⋂ C)
A ⋂ A = A
A ∪ A = A
A⋂ Ø = Ø
U ⋂ A = A
A ∪ Ø = A
U ∪ A = U
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.
- Complement Law A∪A' = U, A⋂A' = Ø and A' = U - A
- De Morgan's Laws (A ∪B)' = A' ⋂B' and (A⋂B)' = A' ∪ B'
- Law of Double complementation (A')' = A
- Laws of Empty set and Universal Set Ø' = ∪ and ∪' = Ø
- A - A = Ø
- B - A = B⋂ A'
- B - A = B - (A⋂B)
- (A - B) = A if A⋂B = Ø
- (A - B) ⋂ C = (A⋂ C) - (B⋂C)
- A ΔB = (A-B) U (B- A)
- n(AUB) = n(A - B) + n(B - A) + n(A⋂B)
- n(A - B) = n(A∪B) - n(B)
- n(A - B) = n(A) - n(A⋂B)
- n(A') = n(∪) - n(A)
Related Reads
Question 1: What is a set in mathematics? Give examples.
Solution:
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?
Solution:
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?
Solution:
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}.
Solution:
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.
Solution:
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?
Solution:
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?
Solution:
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 do not play any of the three games. Find
- Number of students who play badminton, table tennis, and rugby.
- Number of students who play badminton and not rugby.
- Number of students who play badminton and rugby and not table tennis.
Solution:
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
- Number of students who play badminton, table tennis and rugby = 11.
- Number of students who play badminton but not rugby, n(B-R) = n(B)-n(B∩R)=92-16=76
- 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.
Solution:
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.
Conclusion
The Set theory is fundamental in mathematics providing a structured way to the discuss and analyze collections of objects. The key formulas including those for the union, intersection and differences of the sets enable precise calculation and manipulation of these collections. The Mastery of set theory formulas is crucial for the solving complex problems in the various mathematical fields and real-world applications from the computer science to probability theory.
Similar Reads
Set Theory Set theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
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.Example: A = { 2, 4, 6, 8 }. A is a set and 2, 4
8 min read
Representation of Set
Types of Set
Types Of SetsIn mathematics, a set is defined as a well-defined collection of distinct elements that share a common property. These elements— like numbers, letters, or even other sets are listed in curly brackets "{ }" and represented by capital letters. For example, a set can include days of the week. The diffe
13 min read
Empty SetEmpty Sets are sets with no items or elements in them. They are also called null sets. The symbol (phi) ∅represents the empty set and is written as ∅ = { }. It is also known as a void set or a null set. When compared to other sets, empty sets are seen to be distinctive.Empty sets are used to simplif
10 min read
Disjoint SetsDisjoint Sets are one of the types of many pair of sets, which are used in Set Theory, other than this other types are equivalent sets, equal sets, etc. Set Theory is the branch of mathematics that deals with the collection of objects and generalized various properties for these collections of objec
8 min read
Finite SetsFinite set is a collection of finite, well-defined elements. For better understanding, imagine you have a bunch of your favourite toys or snacks. You know exactly how many you have, that's the idea of a finite set in math. A finite set is a way to discuss collections of things you can count. In this
10 min read
Universal SetsUniversal Set is a set that has all the elements associated with a given set, without any repetition. Suppose we have two sets P = {1, 3, 5} and Q = {2, 4, 6} then the universal set of P and Q is U = {1, 2, 3, 4, 5, 6}. We generally use U to denote universal sets. Universal Set is a type of set that
6 min read
Subsets in MathsSubsets in Maths are a core concept in the study of Set Theory. It can be defined as a group of elements, objects, or members enclosed in curly braces, such as {x, y, z} is called a Set, where each member of the set is unique and is taken from another set called the Parent Set.This article explores
12 min read
Operation on Sets
Set OperationsA set is simply a collection of distinct objects. These objects can be numbers, letters, or even people—anything! We denote a set using curly brackets.For example: A = {1, 2, 3}Set Operations can be defined as the operations performed on two or more sets to obtain a single set containing a combinati
10 min read
Union of SetsUnion of two sets means finding a set containing all the values in both sets. It is denoted using the symbol '∪' and is read as the union. Example 1:If A = {1, 3. 5. 7} and B = {1, 2, 3} then A∪B is read as A union B and its value is,A∪B = {1, 2, 3, 5, 7}Example 2:If A = {1, 3. 5.7} and B = {2, 4} t
12 min read
Intersection of SetsIntersection of Sets is the operation in set theory and is applied between two or more sets. It result in the output as all the elements which are common in all the sets under consideration. For example, The  intersection of sets A and B is the set of all elements which are common to both A and B.In
11 min read
Difference of SetsDifference of Sets is the operation defined on sets, just like we can perform arithmetic operations on numbers in mathematics. Other than the difference, we can also perform the union and intersection of sets for any given set. These operations have a lot of important applications in mathematical pr
10 min read
Complement of a SetIn mathematics, a set is a collection or grouping of well-defined objects. All such objects, when grouped in a set, are called elements. Sets are represented by capital letter symbols, and the elements are placed together in a curly bracket {}.For example, if W is the set of whole numbers, then W =
10 min read
Cartesian Product of Sets The term 'product' mathematically refers to the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5.To understand the Cartesian product of sets, one must first be familiar with basic set operations such as union and intersection, which are appli
7 min read
Application of Set
De Morgan's Law - Theorem, Proofs, Formula & ExamplesDe Morgan's law is the law that gives the relation between union, intersection, and complements in set theory. In Boolean algebra, it gives the relation between AND, OR, and complements of the variable, and in logic, it gives the relation between AND, OR, or Negation of the statement. With the help
13 min read