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# Relations

Often in maths, we come across with the word ‘relation’. Generally speaking, by relation we usually understand some connection between the two living or non-living things. Like the relations of mother-daughter, brother-sister, teacher-student etc. We are quite familiar with these relations. Today, we will learn about a new concept of “relations” in maths. Mathematically, we can also define a relationship between the two elements of a set. Let’s begin!

### Suggested Videos        Cartesian Product Introduction to Relations Functions ## Relation

Given any two non-empty sets A and B, A relation R from A to B is a subset of the Cartesian product A x B and is derived by describing a relationship between the first element (say x) and the other element (say y) of the ordered pairs in A & B.

Consider an example of two sets,  A = {2, 5, 7, 8, 9, 10, 13} and B = {1, 2, 3, 4, 5}. The Cartesian product A × B has 30 ordered pairs such as A × B = {(2, 3), (2, 5)…(10, 12)}. From this, we can obtain a subset of A × B, by introducing a relation R between the first element and the second element of the ordered pair (x, y) as

R = {(x, y): x = 4y – 3, x ∈ A and y ∈ B}

Then, R = {(5, 2), (9, 3), (13, 4)}. (Arrow representation of the Relation R)

## Representation of Relation

A relation is represented either by Roster method or by Set-builder method. Consider an example of two sets A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5}. The relation is that the elements of A are the square of the elements of B.

• In set-builder form, R = {(x, y): x is the square of y, x ∈ A and y ∈ B}.
• In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}. ## Terminologies

• Before getting into details, let us get familiar with a few terms:
• Image: Suppose we are looking in a mirror. What do we see? An image or reflection. Similarly, for any ordered pairs, in any Cartesian product (say A × B), the second element is called the image of the first element.
• Domain: The set of all first elements of the ordered pairs in a relation R from a set A to a set B.
• Range: The set of all second elements in a relation R from a set A to a set B.
• Codomain: The whole set B. Range ⊆ Codomain.

## Total Number of Relations

For two non-empty set, A and B. If the number of elements in A is h i.e., n(A) = h & that of B is k i.e., n(B) = k, then the number of ordered pair in the Cartesian product will be n(A × B) = hk. The total number of relations is 2hk.

## Question 1: Let A = {5, 6, 7, 8, 9, 10} and B = {7, 8, 9, 10, 11, 13}. Define a relation R from A to B by

R = {(x, y): y = x + 2}. Write down the domain, codomain and range of R.

Answer :Here, R = {(5, 7), (6, 8), (7, 9), (8, 10), (9, 11)}. Domain = {5, 6, 7, 8, 9}

Range = {7, 8, 9, 10, 11}

Co-domain = {7, 8, 9, 10, 11, 13}.

Question 2: What is the relation in math?

Answer: It refers to a relationship between sets of values. In addition, in mathematics, the relationship is between the x-values and y-values of ordered pairs. Moreover, the set of x-values is known as the domain, and the set of all y-values is known as the range. Furthermore, the brackets show that the values form a set.

Question 3: What makes a relation a function?

Answer: Generally, relationship refers to a set X to a set Y is called a function of each element of X is related to exactly one element in Y. As given an element x in X, there is only one element in Y that is related to then this is a function as each element from X is related to only one element in Y.

Question 4: What are the properties of relation?

Answer: The properties of relation are reflexivity, transitivity, symmetry, and connectedness. Besides, we consider some properties of binary relations.

Question 5: Are relations sets?

Answer: A binary relation over two sets X and Y is a set of ordered pairs (x, y) that consist of elements x in X and y in Y. As if it is a subset of the Cartesian product X × Y. Most importantly, it encodes the information of relation.

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