Question:
$R $ is a relation on $N$ given by $R =\{(x, y ) | 4x +3y = 20\}$. Which of the following belongs to $R$ ?
$R $ is a relation on $N$ given by $R =\{(x, y ) | 4x +3y = 20\}$. Which of the following belongs to $R$ ?
Updated On: May 6, 2024
- (-4, 12)
- (5, 0)
- (3, 4)
- (2, 4)
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The Correct Option is D
Solution and Explanation
Given, $R=\{(x, y): 4 x+3 y=20\}$
Since, $R$ is a relation on $N$, therefore $x , y$ are the elements of $N$. In the given options $(a), (b)$ and $(d)$ satisfy the given relation, but in options $(a)$ and $(b)$ elements are not natural numbers.
Hence, option $(d)$ is the correct answer.
Since, $R$ is a relation on $N$, therefore $x , y$ are the elements of $N$. In the given options $(a), (b)$ and $(d)$ satisfy the given relation, but in options $(a)$ and $(b)$ elements are not natural numbers.
Hence, option $(d)$ is the correct answer.
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Concepts Used:
Relations
A relation in mathematics defines the relationship between two different sets of information. If two sets are considered, the relation between them will be established if there is a connection between the elements of two or more non-empty sets. Therefore, we can say, ‘A set of ordered pairs is defined as a relation.’
Read Also: Relation and Function
Types of Relations:
There are 8 main types of relations which are:
- Empty Relation - An empty relation is one in which there is no relation between any elements of a set.
- Universal Relation - A universal is a type of relation in which every element of a set is related to each other. Now one of the universal relations will be R = {x, y} where, |x – y| ≥ 0. For universal relation, R = A × A
- Identity Relation - In an identity relation, every element of a set is related to itself only. For example, in a set A = {a, b, c}, the identity relation will be I = {a, a}, {b, b}, {c, c}.
- Inverse Relation - It is seen when a set has elements which are inverse pairs of another set. For example if set A = {(a, b), (c, d)}, then inverse relation will be R-1 = {(b, a), (d, c)}.
- Reflexive Relation - If every element of set A maps for itself, then set A is known as a reflexive relation.It is represented as a∈ A, (a,a) ∈ R.
- Symmetric Relation - A relation R on a set A is known as asymmetric relation if (a, b) ∈R then (b, a) ∈R , such that for all a and b ∈A.
- Transitive Relation - For transitive relation, if (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R. For a transitive relation, aRb and bRc ⇒ aRc ∀ a, b, c ∈ A
- Equivalence Relation - If a relation is reflexive, symmetric and transitive at the same time it is known as an equivalence relation.
Representation of Relations:
There are two ways by which a relation can be represented-
- Roster method
- Set-builder method
The roster form and set-builder for for a set integers lying between -2 and 3 will be-
Roster form
I= {-1,0,1,2}
Set-builder form
I= {x:x∈I,-2<x<3}