Smallest Equivalence Relation on Real Numbers: Proving with Line y-x=1

[dharper8861]

1) Recall that an equivalence relation S on set R ( R being the reals) is a subset of R x R such that

(a) For every x belonging to R (x,x) belongs to S
(b) If (x,y) belongs to S, then (y,x) belongs to S
(c) If (x,y) belongs to S and (y,z) belongs to S then (x,z) belongs to S

What is the smallest equivalence relation S on the Set R of real numbers that contains all the points in the line y - x = 1. Prove your answer.

Can anyone help figure this out? I am pretty lost on this one.

 

[LCKurtz]

Have you tried drawing a picture of S? You know it has at least the line y - x = 1. What does (a) tell you in terms of your picture? And (b)? Does (c) work for your picture?