Having some problems with this question and hoping someone could help.
Let $S$ and S' be the following subsets of the plane: $S = \{(x,y): y=x+1\text{ and }x\text{ a member of }(0,2)\},$ S'= \{(x,y): y-x\text{ is an integer}\}.
A. Show that S' is an equivalence relation on the real line and $S$ is a subset of S'. Describe the equivalence classes of S'.
B. Show that given any collection of equivalence relations on a set $A$, their intersection is an equivalence relation on $A$.
C. Describe the equivalence relation $T$ on the real line that is the intersection of all equivalence relations on the real line that contain $S$. Describe the equivalence classes of $T$.
C is the problem I am having the most difficulty with.
Answers so far:
a.
- EDIT: Symmetric: if (x,y) works then, $y-x=n \rightarrow x-y=-n$
Reflexive because $x-x = 0$
EDIT: Transitive $x-y=n$ and $y-z=k$ $\rightarrow y=k+z \rightarrow x-(k+z)=n \rightarrow x-z = k+n=p$
$S: y=x+1 \rightarrow y-x=1$, 1 integer out of the set of all integers in S'.
Equivalence classes of $S$ would be all diagonal lines with slope 1 through $y=n$ and $x=n$.
b. $E_1$ and $E_2$ equivalence relations. Intersections both contain $(x,y)$ and $(y,x)$ because if they are members of either $E_1$ or $E_2$ they are satisfy equivalence requirements.
c. I am not sure how to answer this. I thought the intersection of all equivalence relations on the real line containing $S$ would be $S$.
Any help would be greatly appreciated.