What is an equivalence relation?
Show that the relation of'similarity' on the set S of all triangle in a plane is an equivalence relation.

Explanation

Let A be the set of triangles in a plane and relation
R = `{(Delta_(1), Delta_(2)) : Delta_(1), Delta_(2)in A and Delta_(1)~ Delta_(2)}`
(i)Let `Delta in A`
`:. ""_(Delta)R_(Delta) rArrDelta ~ Delta`
`:'` Every triangle is similar to itself,
`:. _(Delta)R_(Delta)` is true.
`rArr` R is reflexive.
(ii) Let `Delta_(1), Delta_(2)in A and ""_(Delta_(1))R_(Delta_(2))`
Now, `""_(Delta_(1))R_(Delta_(2)) rArr Delta_(1)~ Delta_(2)`
`rArrDelta_(2)~Delta_(1)`
` rArr ""_(Delta_(2))R_(Delta_(1))`
`:.` R is symmetric.
(iii) Let `Delta_(1), Delta_(2), Delta_(3)in A and _(Delta_(1))R_(Delta_(2)) and _(Delta_(2))R_(Delta_(3))`
Now, `_(Delta_(1))R_(Delta_(2))and _(Delta_(2))R_(Delta_(3))`
`rArr ""Delta_(1) ~ Delta_(2) and Delta_(2)~Delta_(3)`
`rArr ""Delta_(1) ~ Delta_(3)`
`rArr ""_(Delta_(1))R_(Delta_(3))`
`:.` R is transitive.
`:.` R is reflexive, symmetric and transitive.
`rArr` R is an equivalence relation. Hence proved.