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 $R$ .
Last updated date: 18th Mar 2023
•
Total views: 303.6k
•
Views today: 3.85k
Answer
303.6k+ views
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.
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.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?
