InterviewSolution
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InterviewSolution
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Let T be the set of all triangles in a plane with R a relation in T given by `R={(T_1,T_2): T_1( i s c o n g r u e n t t o T)_2}`. Show that R is an equivalence relation. |
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Answer» Since a relation `R` in `T` is said to be an equivalence relation if `R` is reflexive, symmetric and transitive. (i) Since every triangle is congruent to itself `:.R` is reflexive (ii) `(T_(1),T_(2))epsilonRimpliesT_(1)` is congruent to `T_(2)impliesT_(2)` is congruent to `T_(1)implies(T_(2),T_(1))epsilonR` Hence `R` is symmetric. (iii) Let `(T_(1),T_(2))epsilonR ` and `(T_(2),T_(3))epsilonimpliesT_(1)` is congruent to `T_(2)` and `T_(2)` is congruent to `T_(3)` `implies T_(1)` is congruent to `T_(3)implies(T_(1),T_(3))epsilonR` `:.R` is transitive Hence `R` is an equivalence relation. |
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