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.

Answer 1

Hint:-For solving these questions, we would be requiring knowledge about symmetric, reflexive and transitive functions.

Complete step-by-step answer:
A relation is a relationship between sets of values. In math, the relation is between the x-values and y-values of ordered pairs. The set of all x-values is called the domain, and the set of all y-values is called the range.
The types of relations are as follows
Reflexive Relation
A relation is a reflexive relation if every element of set A maps to itself. i.e. for every a \[\in \]
 A, (a, a) \[\in \] R.
Symmetric Relation
A symmetric relation is a relation R on a set A if (a, b) \[\in \] R then (b, a) \[\in \] R, for all a and b \[\in \] A.
Transitive Relation
If (a, b) \[\in \] R, (b, c) \[\in \] R, then (a, c) \[\in \] R, for all a, b, c \[\in \] A and this relation in set A is transitive.
Equivalence Relation
If and only if a relation is reflexive, symmetric and transitive, it is called an equivalence relation.
As mentioned in the question, we have to show that the given relation is an equivalence relation.
Now, we can define the function as follows
 R= {(P, Q): OP=OQ} for O being the origin.
For checking the reflexivity, we can write the following
Now if we take point P, we can write as follows
OP=OP
(Because the distance of point P from origin would be the same)
⇒ (P, P) \[\in \] R.
Hence, we can say that R is reflexive.
 For checking the symmetry, we can write the following
Now, if we take two different points P and Q such that we can write the following
OP=OQ
⇒OQ=OP
(Because they are the two sides of an equation and we can flip their sides without making any change in their values)
⇒ (P, Q) \[\in \] R and (Q, P) \[\in \] R.
Hence, we can say that R is symmetric.

For checking the transitivity, we can write the following
Now, if we take three different points P and Q and R such that we can write the following
OP=OQ and OQ=OR
⇒OQ=OP=OR
⇒OP=OR
⇒​ (P, Q) \[\in \] R, (Q, R) \[\in \] R and (P, R) \[\in \] R.
Therefore, we can say that R is transitive.
Hence, the given relation is an equivalence relation.

Note:-The students can make an error if they don’t know about the definitions and the meaning of different types of relations.
Also, it is important to know about the different types of set representations that are the rooster form or the set builder form as without knowing these one could never understand the question properly.