Let $R =\{( P , Q ) | P$ and $Q$ are at the same distance from the origin $\}$ be a relation, then the equivalence class of (1,-1) is the set:
- $S=\left\{(x, y) | x^{2}+y^{2}=4\right\}$
- $S =\left\{( x , y )| x ^{2}+ y ^{2}=1\right\}$
- $S =\left\{( x , y ) | x ^{2}+ y ^{2}=\sqrt{2}\right\}$
- $S =\left\{( x , y ) \mid x ^{2}+ y ^{2}=2\right\}$
The Correct Option is D
Solution and Explanation
To solve the given problem, we need to determine the equivalence class of the point \((1, -1)\) using the relation \(R =\{( P , Q ) | P\) and \(Q\) are at the same distance from the origin \(\}\).
- First, we calculate the distance of the point \((1, -1)\) from the origin. The distance from the origin is given by the formula: \(d = \sqrt{x^2 + y^2}\). For the point \((1, -1)\):
\(d = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}\)
- The equivalence class of a point under this relation consists of all points that are equidistant from the origin as the given point. Therefore, we need all points \((x, y)\) such that the distance from the origin is \(\sqrt{2}\):
\(\sqrt{x^2 + y^2} = \sqrt{2}\)
- Squaring both sides, we obtain:
\(x^2 + y^2 = 2\)
- Thus, the set of all points \((x, y)\) satisfying this equation is:
\(S = \{(x, y) \mid x^2 + y^2 = 2\}\)
Therefore, the equivalence class of the point \((1, -1)\) is the set of points \((x, y)\) such that \(x^2 + y^2 = 2\). This corresponds to the correct answer:
\(S = \{(x, y) \mid x^2 + y^2 = 2\}\)
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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}