Sets Formula: Set Theory, Properties, Types & Examples

Sets Formula is a list of fundamental formulas related to Set Theory in Mathematics. Set is defined as a well-organized collection of objects and elements in Maths. 

• Sets Formulas are used in numerous fields such as Statistics, Probability, Geometry, and Sequences.
• They are used in various set operations such as union, intersection, complement, and difference of sets.
• Venn Diagrams are used to visualize set formulas and provide their proof. 

Set Theory is a branch of mathematical logic that studies Sets, their properties, operations, and symbols.

Read More: NCERT Solutions for Class 11 Maths Sets

Key Terms: Sets Formula, Set Theory, Sets, Union of Sets, Complement of Set, Set Operations, Commutative Law, Universal Set

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What is Set Theory?

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Set Theory is a branch of Mathematics that deals with Sets. It is widely used in the mathematical classification and organization of data. 

• Set Theory was formulated by Georg Cantor, a German mathematician.
• Set is defined as an organized collection of elements or objects.  
• Elements are the items present in a set and can be numbers, alphabets, variables, etc.
• Sets are denoted by a capital letter and their elements are listed within curly brackets like A = {...}. 

Example: Consider a Set of First Five Odd Numbers. It will be expressed as follows:

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What is Sets Formula?

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Sets Formula is a list of standard mathematical formulas derived from Set Theory. They are used in various set operations: 

If there are two sets, Set A and Set B, then:

• The number of items in each set A or B is represented by the expression n (A U B).
• The number of elements present in both sets A and B is shown by the expression n (A ⋂ B).
• n (A U B) = n(A) + (n(B) - n (A ⋂ B)

If A, B, and C are the three sets in Universal Set U, then,

Set Detailed Video Explanation

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Complement of Sets Formula

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Sets Formula for the Complement of Sets includes the Basic Complement Law, De Morgan's Laws, Double Complement, Laws of Empty Sets 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

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Difference of Sets Formula

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Sets Formulas for the difference of sets across two sets, across a null set, and for the complement of a set are as follows:

• 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(A∪B) = n(A) + n(B) – n(A⋂B)
• n(A∪B∪C) = n(A) +n(B) + n(C) – n(B⋂C) – n (A⋂ B) – n (A⋂C) + n(A⋂B⋂C)
• n (A – B) = n(A∪B) – n(B)
• n (A – B) = n(A) – n(A⋂B)
• n(A’) = n (∪) – n(A)

Other Important Sets Formula

Some other important sets formulas are as follows: 

• n(U) = n (A) + n(B) + – n(A⋂B) + n((A∪B)’)
• n((A∪B)’) = n(U) + n(A⋂B) – n(A) – n(B)

Read More: ​Sets Important Questions

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Sets Formula Solved Examples

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Given below are a few solved examples on Sets Formula for a better understanding of the formulas: 

Example 1: In a playground, each student plays either cricket or badminton, or both. The number of students who play cricket or badminton or both is 11, 12 and 3 respectively. Find out how many students play either cricket or badminton.

Solution: Let us assume that 

• n(Cricket)= n(P)
• n(Badminton) = n(Q)
• n(Either Cricket or Badminton) = n(P∪Q)
• n(Cricket and Badminton) = n(P∩Q)

It is given that, n(P) = 12 , n(Q) = 12 and (P∩Q) = 3

Using the Sets Formula, we get

n(P∪Q) = n(P) + n(Q) - n(P∩Q) = 11 + 12 - 3 = 20

Thus, the students that play either cricket or badminton are 20.

Example 2: There are 100 students in a Hobby Club of which 35 like drawing and 45 like music. 10 students like both drawing and music. What will the number of students that like either of them or neither of them?

Solution: According to the question, 

• Total Number of Students = 100
• Number of Students who like Painting, n(P) = 35
• Number of Students who like Dancing, n(D) = 45
• Number of Students who like Both, n(P∩D) = 10

We need to find the number of students who like either of them (n(P ∪ D)) and the number of students who like neither (n(P∪D)).

Using Sets Formula,

n(P∪D) = n(P) + n(D) – n(P∩D)

45 + 35 - 10 = 70

Thus, the number of students who like either music or drawing is 70.

Number of Students Who like Neither = Total Students – n(P∪D) = 100 – 70 = 30 

Thus, 30 students like neither music nor drawing.

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Properties of Sets

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The properties of Sets are very similar to those of real numbers and natural numbers. Sets also abide by Distributive Law, Associative Law, and Commutative Law. The important properties of Sets are listed as follows: 

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Types of Sets

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Sets are divided into various categories depending on the types of elements included in them. The various types of sets are listed below: 

Finite Set

• A finite set is one that has a finite or countable number of elements.
• For instance, Set B = {k | k is a prime number smaller than 20}
• Hence, B = {2, 3, 5, 7, 11, 13, 17, 19}

Infinite Set

• An infinite set is one that has an unlimited number of elements.
• For instance, Set C is a set of "Multiples of 3".
• Hence,  C = {3, 6, 9, 12, 15……..}.

Empty or Null Set

• An empty set or null set is a set that is devoid of any elements, i.e. it doesn’t have any elements.
• An empty set is expressed with the symbol '∅'. It is pronounced as "phi."
• Example: Assume Set X = {}.

Singleton Set

• A singleton set, also known as a unit set, is a set that consists of just one element.
• For example, A = {k | k is an integer between 3 and 5}.
• Hence, A = {4}. 

Equal Sets

• Equal Sets are those sets that include the same elements in both sets.
• For instance, A = {1, 2, 3}, and B = {1, 2, 3{.
• In this instance, sets A and B are equal sets. A = B is used to symbolize this.

Unequal Sets

• Two sets are considered unequal if they each include at least one unique element.
• For instance, A = {1, 2, 3}, and B = {2, 3, 4}.
• In this instance, sets A and B are not equal. A ≠ B is used to illustrate this.

Equivalent Sets

• When two sets have the same number of items, even though the elements are different, they are said to be equivalent sets.
• For instance, A = {1, 2, 3, 4}, and B = {a, b, c, d}.
• Since n(A) = n, set A and set B here are equivalent sets.

Overlapping Sets

• If there is at least one element from set A in set B, then the two sets are said to overlap.
• For instance, A = {2, 4, 6} and B = {4,8,10}.
• In this case, element 4 is present in both sets A and B. A and B are thus overlapping sets.

Disjoint Sets

• If there are no shared elements between two sets, they are disjoint sets.
• Consider A = {1, 2, 3, 4} and B = {5, 6, 7, 8}.
• Here, Sets A and B are disjoint sets as they have all different elements. 

Superset and Subset

• If every element of one set is present in another, the first set is said to be a subset of the second one (A ⊆ B).
• Also, the second set will be the superset of the first set (B ⊇ A).
• Consider, A = {4,8,6} and B = {0,1,2,3,4,5,6,7,8,9}
• A ⊆ B as all the elements in set A are present in set B. So, A is the subset of B.
• Thus, B ⊇ A denotes that set B is the superset of set A.

Power Set

• Power set is defined as a set of all the subsets that a set could contain.
• Cosnider, Set A = {3,4}.
• Power Set of A i= {{∅}, {3}, {4}, {3,4}}.

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Things to Remember

• Sets Formula are the fundamental formulas used in Set Theory in Mathematics.
• They are derived from Set Theory and are used in various set operations. 
• Set Theory is a mathematical theory that deals with Sets and their related operations.
• Set is a clearly defined group of elements or items in Maths. 
• Elements are the items that make up a set and can be any number, alphabet, or object.
• A set is expressed by a Capital letter and its elements are added within a curly bracket like B = {….}.
• Singleton Sets, Finite Sets, Infinite Sets, Empty Sets, Equal Sets, etc. are all types of sets. 
• Union, Intersection, Complement, Difference, and Cartesian Product are various operations on sets.

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Previous Years’ Questions

1. If a,b and c∈N which one of the following is not true… [KCET – 2006]
2. A class has 175 students. The following students… [BITSAT – 2013]
3. If A and B are finite sets and… [KCET – 2017]
4. Write the set builder form , A = -1… [KCET – 2015]
5. A subset of the additive group of real numbers… [KCET – 2006]
6. If A = {1,2,3,4,5,6}, then the number of subsets… [KCET – 2020]
7. In a certain town 25% families own a cell phone… [KEAM]
8. In a class of 60 students, 25 students… [KCET – 2014]
9. If A\B = {a, b}, B\A = {c, d} and A\B = {e, f}… [KEAM]
10. If n(A) = 43, n(B) = 51 and… [KEAM]