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Question

Let A be the set of all points in a plane and let O be the origin. Show that the relation {R=(P,Q):P,QA and OP=OQ} is an equivalence relation.


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Solution

Determine the relations:

Let O be the origin in the plane and P,Q and X be three point on the plane

Now,

  1. OP=OP is true always. This means R is reflective.
  2. OP=OQ or OQ=OP. This means R is symmetric.
  3. OP=OQ and OQ=OX. So, OP=OX. This means R is transitive.

Hence, R is an equivalence relation.


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