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Question
Prove that the relation R defined on the set V of all vectors by `vec"a" "R" vec"b"` if `vec"a" = vec"b"` is an equivalence relation on V
Solution
`vec"a" "R" vec"b"` is given as `vec"a" = vec"b"`
(i) `vec"a" = vec"a"`
⇒ `vec"a" "R" vec"a"`
(i.e.,) the relation is reflexive.
(ii) `vec"a" = vec"b"`
⇒ `vec"b" = vec"a"`
(i.e.,) `vec"a" "R" vec"b" - vec"b" "R" vec"a"`
So, the relation is symmetric.
(iii) `vec"a" = vec"b" ; vec"b" = vec"c"`
⇒ `vec"a" = vec"c"`
(i.e.,) `vec"a" "R" vec"b" ; vec"b" "R" vec"c"`
⇒ `vec"a" "R" vec"c"`
So the given relation is transitive
So, it is an equivalence relation.
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