Question

Let $R$ be the relation on $Z$ defined by $R=\left\{ (a,b):a,b\in Z,a-b\text{ is an integer} \right\}$ . Find the domain and range of $R$ .

Answer 1

Hint: We are given that $R$ is the relation on $Z$ defined by $R=\left\{ (a,b):a,b\in Z,a-b\text{ is an integer} \right\}$. So we know that the difference of integers is always integers. Try it, you will get the answer.

Complete step-by-step answer:
Relations and its types of concepts are some of the important topics of set theory. Sets, relations, and functions all three are interlinked topics. Sets denote the collection of ordered elements whereas relations and functions define the operations performed on sets. A relation is said to be asymmetric if it is both antisymmetric and irreflexive or else it is not.
The relations define the connection between the two given sets. Also, there are types of relations stating the connections between the sets.
Sets and relations are interconnected with each other. The relation defines the relation between two given sets.
If there are two sets available, then to check if there is any connection between the two sets, we use relations. In discrete Maths, an asymmetric relation is just opposite to a symmetric relation. In a set A, if one element less than the other satisfies one relation, then the other element is not less than the first one. Hence, less than (<), greater than (>), and minus (-) are examples of asymmetric. We can also say, the ordered pair of set A satisfies the condition of asymmetric only if the reverse of the ordered pair does not satisfy the condition. This makes it different from symmetric relation, where even if the position of the ordered pair is reversed, the condition is satisfied. A relation is said to be asymmetric if it is both antisymmetric and irreflexive or else it is not.
For example, an empty relation denotes none of the elements in the two sets is the same.
It is given in the question that, $a,b\in Z$.
The relation $R$ on $Z$ is defined by

$R=\left\{ (a,b):a,b\in Z,a-b\text{ is an integer} \right\}$
As we know the difference of integers is always integers.
So, Domain $R$$=$ $Z$.
 Also, range $R$$=$ $Z$.

Note: Read the question carefully. Also, you must know the concept behind the relation. You should be familiar with the formulae. Do not make a silly mistake while simplifying. Solve the problem in a step by step manner.