Relations

Definition of Relation in math

Relations, as the word suggests, define the relationship between two sets. Relation in mathematics could be defined as the connection or relationship between the elements of two or more sets, given that the sets are non-empty sets. 

Relations are the operations performed on sets. Relations are formed by the Cartesian product of the subset of two sets depending upon the condition of the relation.

A relation between two or more non-empty sets is established only when there is a connection between the elements of these sets.

For example, consider two sets A and B, such that A= {1, 2, 3, 4, 5, 6} and B= {4, 9, 10, 12, 16, 20}

A relation between these two sets is to be established such that the elements of B are the square of the elements of A.

The Relation R= {(2, 4), (3, 9), (4, 16)}

Terminologies of relations in math

Before understanding the concept of relations in detail, a student should fully get a grip on the common terms used in relations in math.

1) Domain- The domain is defined as the set of all ordered pairs in a relation R.

    In the above example, domain = {2, 3, 4}

2) Range-  The range is defined as a set of all second elements in a relation R from a set A to a set B

In the above example, range = {4, 9, 16}

3) Image- In any ordered pair (a, b), the lament b is called the reflection of the element a

4) Co-domain- The co- domain is the whole set B. 

Representation of Relations in Math

Relations in math could be represented in any of the following three ways:

1) Roster form- Relations could be represented in the roster form. In this form, a set is represented with a list of all its elements that are separated by commas, enclosed within brackets.

For example, R = {(2, 4), (3, 9), (4, 16)} is a relation in mathematics represented in the roster form.

2) Set builder form- In this form, the elements of a relation are described in words, stating properties that satisfy the conditions to be a member of the relation.

For example, R = {(a, b): a belongs to A and b belongs to A, A is the square root of B}

3) By arrow diagram- In this method, the relation between two sets is represented by showing an arrow from one set pointing to the other set, defining a relation between them.

Types of Relations in Math

Relations are used to determine a connection between two or more sets. There are eight types of relations that define the different relationships that could be between the elements of two or more nonempty sets.

1) Empty relation- If no element of a set A is related to any element of the set B, then the relation between the elements of set A and set B is called an empty relation.

An empty relation is also known as a void relation or a trivial relation.

Example- If set A={Hens} and set B = {Pens}, then the relation between them is an example of an empty relation.

2) Universal relation- If all the elements of a set A are related to the elements of set B, then the relation between them is called a universal relation. It is also known as a full relation as every element of set A is in relation with the elements of set B.

Universal relation is also known as a trivial relation.

Example- If set A = {4, 9, 16}, B= {2, 3, 4} and R = {(a, b): a belongs to A and b belongs to A, A is the square root of B}

3) Identity relation- It is a relation where every element of a set A is related to itself. Such a relation would be a subset of the set A where a is an element of the set A and (a, a) is a relation of ordered pairs such that both the elements in the relation belong to the same set A.

For example : If A = {1,2,3}, then the relation IA ={(1,1),(2,2),(3,3)} is the identity-relation on set A. 

4) Inverse relation- It is the inverse of a given relation that is obtained by interchanging the elements of the ordered pairs of the relation. The first element becomes the second element and vice versa, changing the values of the domain and range.

For Example, The inverse relation of the relation R= {( 1, 2), ( 9, 0)} will be R^-1= {(2, 1), (0, 9)}.

5) Reflexive relation-  A relation R derived from a set A is reflexive if every element of the relation is related to itself.

For example, consider a set A = {2, 4}. Now, the reflexive relation will be R = {(2, 2), (4, 4), (4, 2), (2, 4)}.

6) Symmetric relation- A relation (a,b) on set A is symmetric if a = b and b = a is true. 

For example, the relation R={(4,5),(5,4),(6,5),(5,6)} on set A={4,5,6} is symmetric as a = b is true and b = a is also true.

7) Transitive relation- A relation R on a set A is transitive if, for all elements m, n, l in X, whenever R relates m to n and n  to l, then R also relates a to c.

For example- A  =  {a, b, c} and R  = {(a, b), (b, c), (a, c)}, then R is a transitive relation.

8) Equivalence relation- A relation R is said to be an equivalence relation when the relation is reflexive, symmetric and transitive.

Conclusion

Relations math is a common question asked in set theory, and a good comprehension of this concept can make a student crack the toughest questions based on set theory. A good understanding of relations math will help students get a good grip on the set theory that usually has a good weightage in competitive exams.