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Question
Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.
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Solution
Let A be the set of all points in a plane such that
A={P : P is a point in the plane}
Let R be the relation such that R={(P, Q) : P, Q∈A and OP=OQ, where O is the origin}
We observe the following properties of R.
Reflexivity: Let P be an arbitrary element of R.
The distance of a point P will remain the same from the origin.
So, OP = OP
⇒ (P, P) ∈ R
So, R is reflexive on A.
Symmetry : Let (P, Q) ∈ R
⇒ OP = OQ
⇒ OQ = OP
⇒ (Q, P) ∈ R
So, R is symmetric on A.
Transitivity: Let (P, Q), (Q, R) ∈ R
⇒ OP= OQ and OQ = OR
⇒ OP= OQ = OR
⇒ OP = OR
⇒ (P, R) ∈ R
So, R is transitive on A.
Hence, R is an equivalence relation on A.
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