SET Theory: Definition, Weightage, Important Formulae, and Previously asked Questions

Set theory is one of the basic and scoring topics asked in the Engineering mathematics section of GATE Exam. A candidate must prepare for this kind of topic well to score good marks. In GATE, 2-3 questions are asked from set, relations, and functions topic making it a 3-4 percent of the whole set of questions. All the important operations, formulae and previous year questions related to set theory are elaborated in this article.

What is a SET?

A set is nothing but a collection of objects. Even the set itself is an object. Sets always contain unique identities. There are different types of sets such as singleton sets, finite sets, infinite sets, subsets, empty sets, equal sets, overlapping sets, and disjoint sets.

• Sets can be represented using diagrams representation known as Venn-diagrams.
• One set also contains another set.
• Mathematical operations like union, intersection, complement, and differences can be performed upon sets.

Major terms and concepts related to set theory

All the possible terminologies related to SETs are mentioned below:-

Principle of Extensionality

The principle of Extensionality states that ‘Two sets A and B can be equal if and only if element x belonging to set A also strictly belongs to set B.’
It can be mathematically represented as:

Set Operations

Different operations such as Union, Intersection, Difference, and Complement can be performed on any number of sets.

Union: A set union contains all the common as well as uncommon elements of sets. This operation is performed between two or more sets.

Intersection: A set intersection contains all the common elements of two or more sets.

Differences: Set differences operation removes the common element from a set and represents the dissimilar one. It is performed between two sets

Complement: A complement is the representation of elements which does not belong to a particular set.

Types of Sets

Singleton set: The set which contains a single element is called a singleton set.
Example: A={3}

Finite set: The set which contains a finite number of elements is called a finite set.
Example: A= {Set of first five natural numbers} = {1,2,3,4,5}

Infinite set: The set which contain an infinite number of elements is called infinite set.
Example: A= {Set of natural numbers} = {1,2,3,4,5, ….}

Subsets: Subset of a set is a set that contains all or some of the elements is called subset. An empty set is subset of all sets.
Example: Super set S={1,2,3,4,5,6,7,8,9} is a super set and Set A={1,2,3} is its subset.

Supersets: Superset is a kind of universal set that contains all elements of a subset along with some added aspects in most cases.
Example: Super set S={1,2,3,4,5,6,7,8,9} is a super set and Set A={1,2,3} is its subset.

Empty sets: A set that does not contain any element is called an empty set.
Example: A={}

Equal sets: Two sets are said to be equal sets if they have some elements in each other
Example: A={1,2,3} and B={1,2,3} are equal.

Equivalent Sets: two sets are said to be equivalent sets if they contain the same number of elements.
Example: A={1,2,3} and B={3,4,5} are equivalent sets

Overlapping sets: Two sets are said to be overlapping sets if they contain some or even a single element in common.
Example: A={1,2,3} and B={2,3,4} are overlapping sets.

Disjoint sets: Two sets are said to be disjoint sets if they do not contain even a single element in common.
Example: A={1,2,3} and B={4,5,6} are disjoint sets

Some more terminologies

Venn-Diagrams: They are the way of representing one or more sets in the form of a diagram.

Cardinality: The cardinality represents the total number of elements in the set
Example: A= {1,2,3,4,5}
Cardinality of set, n(A) = 5

Engineering Mathematics Section Weightage in GATE

Graduate Aptitude Test in Engineering is an examination that opens the door to complete your M. Tech from the premium institutions of the country. Engineering mathematics is a major subject asked in the GATE exam. This subject covers almost 15% of the total syllabus. From the vision of a GATE aspirant, Engineering Mathematics can never be ignored. The topic-wise weightage of the Engineering mathematics syllabus is tabulated below:

GATE 2022 Engineering mathematics topic-wise weightage

In GATE 2022, most of the GATE exam sets contained 1-2 questions from the set theory itself. This topic alone covered 3-4 percent of the GATE question set.

Engineering Mathematics SETS: Important formulae and concepts

Some major formulas and concepts to remember from set theory are:

Laws for set theory

Some of the laws that hole for set theory are:

Identity Law
The U and φ are the identity elements for intersection and union respectively

Domination Law
The U dominates the operations of the union and φ dominates the operation of the intersection.
A U U= U
A ∩ φ = φ

Idempotent law
The Idempotent law states that:
A U A= A
A ∩ A= A

Double Negation Law
The complement of complement is the original set.
U – (U-A)= A

Commutative law
The commutative law states that both the operations union and intersection are commutative.
A U B= B U A
A ∩ B= B ∩ A

Associative Laws
The associative law states that both the operations union and intersection are associative.
(A U B) U C = A U (B U C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive Laws
The distributive law states that Union distributes over an intersection and an intersection distributes over a union.
(A ∩ B) U C = A ∩ (B U C)
(A U B) ∩ C = A U (B ∩ C)

De-Morgans Law
De-Morgan’s Law states:
(A U B)’ = (A’ U B’)
(A ∩ B)’ =(A’ ∩ B’)

Negation Laws
The Negation law states that:
A U (U – A) = U
A ∩ (U – A) = φ

Engineering Mathematics SETS: Doubt Areas

Power Sets -> Power sets are the most confused topic of Set theory. Power set of a set A is defined as a set of all subsets of A including A and null set itself. Formula to calculate power set of a set A is:
A= {x, y, z}
Cardinality of A = n(A) = 3
Power set = 2n(A) = 23 = 8
P(A) ={ {} , {x, y, z}, {x, z}, {x, y}, {y, z}, {x}, {y}, {z} }