# Question on equivalence relation.

1. Feb 14, 2007

### MathematicalPhysicist

let R be a reflexive and symmetric relation on a set X. let's define a relation S on X. by (x,y) in S iff there exists a finite sequence x0,x1,...,x_n
of terms from X such that x0=x xn=y and (x_i,x_i+1) in R for i=0,1,...n-1.
now ive proven that S is an equivalence relation, i need to show that S is the smallest equivalence relation that includes R, i.e if R is a subset of T then S is a subset of T.
i think that the naswer is simple but perhaps too simple to be true.
if (x,y) in S, then there exists a finite sequence x,y such that (x,y) in R and so (x,y) in T.
am i right here?