Question

Define a relation on a set. What do you mean by the domain and range of relation? Give an example.

Answer 1

In this question, we simply need to define and explain a relation and its domain and range with respect to a set. A relation is simply a rule that gives or contains sets of ordered pairs as its elements under the relation. It is denoted by R.

Complete step by step solution: Now, first of all the word Relation as is clear means an establishment of a rule, between two clusters or sets of any commodities or quantities. Domain and Range are the characteristic properties, which are extensions to the definition of relation.
Now for example let us consider two sets.
Set A contains the elements 1,2,3,4,5 and 6 such that it is written as:
And there exists another set B which contains the elements:
Let R be a relation defined from A to B. defined as $R:A \to B$such that the elements of the relation set R are
\left\\{ {({x_1},f({x_1})),({x_2},f({x_2})),({x_3},f({x_3}))..........} \right\\} \\\ = \left\\{ {(1,2),(2,3),(3,4),(4,5),(5,7),(6,9)} \right\\} \\\
Hence, a relation from a set A to a set B is defined as $aRb$ that is a collection of ordered pairs, whose x-coordinates constitute the domain and the y-coordinates constitute the range.
Now, by Domain of a relation we mean the elements of the first set that have their images in the second set, which make the x-coordinates of the ordered pairs of the relation R. While Range refers to the images of the elements in the set which make up the y-coordinates of R.
Now, with respect to the sets A and B taken in the example, the set A is the Domain and the Set B is the range. So we can say that for every element x \in \left\\{ A \right\\}, the set A will be the domain. Now for every x \in \left\\{ A \right\\}There is an image $f(x)$ that exists in set B. Hence the set B containing all the images $f(x)$ of the elements x \in \left\\{ A \right\\}. Therefore, set B is called the Range of the Relation R and set A is called the Domain of the relation R.
Hence domain D of the relation R is \left\\{ {1,2,3,4,5,6} \right\\}while the range R is \left\\{ {2,3,4,5,7,9} \right\\},
So for any set A, a relation R is a set of elements which relates the elements of the first set with its corresponding image in the second set, such that the first set containing the elements is called the Domain, and the second set which contains the image of the elements of the first set is called the range.
However, each element of the first set that is Domain, may have more than one image also in the range. Hence it’s not necessary that every element in Domain will have a unique image in its Range.

Note: Relation is not to be confused with a function. A function is a special kind of relation, in which every element of the first set has one and only one image in the second set. For example if we have two sets say, A and B, such that for all elements belonging to the set A, $f$ is a function from A to B, then function $f$, in one-to-one function if every element of A has exactly one unique image in B.