If \(A = {1, 2, 3, … 100}\), \(R = {(x, y) | 2x = 3y, x, y ∈ A}\) is symmetric relation on \(A\) and the number of elements in \(R\) is \(n\), the smallest integer value of \(n\) is
Correct Answer: 0
Solution and Explanation
Top Questions on Relations
- If $A=\{1,2,3,4\}$. A relation from set $A$ to $A$ is defined as $(a,b)\,R\,(c,d)$ such that $2a+3b=3c+4d$. Find the number of elements in the relation.
- Let the set of all relations $ R $ on the set $ \{a, b, c, d, e, f\} $, such that $ R $ is reflexive and symmetric, and $ R $ contains exactly 10 elements, be denoted by $ S $. Then the number of elements in $ S $ is ___.
- The number of relations on the set $ A = \{1, 2, 3\} $ containing at most 6 elements including $ (1, 2) $, which are reflexive and transitive but not symmetric, is:
- Let $ A = \{-3, -2, -1, 0, 1, 2, 3\} $ and $ R $ be a relation on $ A $ defined by $ xRy $ if and only if $ 2x - y \in \{0, 1\} $. Let $ l $ be the number of elements in $ R $. Let $ m $ and $ n $ be the minimum number of elements required to be added in $ R $ to make it reflexive and symmetric relations, respectively. Then $ l + m + n $ is equal to:
- Define a relation \( R \) on the interval \( [0, \frac{\pi}{2}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1 \). Then \( R \) is:
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A substance 'X' (1.5 g) dissolved in 150 g of a solvent 'Y' (molar mass = 300 g mol$^{-1}$) led to an elevation of the boiling point by 0.5 K. The relative lowering in the vapour pressure of the solvent 'Y' is $____________ \(\times 10^{-2}\). (nearest integer)
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- Solutions
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}