Assume real numbers R for now. Consider relation on R, x y iff x y Z. (a) Is it an equivalence relation? (b) Compute [1/2] (c) Classify all elements of R/ . That is state: “Every element of R/ is of the form ...” – include all the details, so that you don’t count any element twice. (d) Bonus question: (you don’t need to answer): Is there a geometric way to think about R/ ? Solution a) x-x=0 is in Z for all real numbers x SO R is reflexive IF, x-y is in Z then y-x is also in Z So, R is symmetrix IF, x-y and y-z are in Z then x-y+y-z=x-z is also in Z Hence, R is transitive Hence, R is an equivalence relation b) [1/2]={n+1/2: n is in Z} c) Elements of R/~ are of the form: a+R, where, a is a real number in (0,1] We need only consider real numbers outside this interval because all other real numbers will differ by an integer from one of the numbers in this interval..

1. Assume real numbers R for now. Consider relation on R, x y iff x y Z.
(a) Is it an equivalence relation?
(b) Compute [1/2]
(c) Classify all elements of R/ . That is state: “Every element of R/ is of the form ...” – include
all the details, so that you don’t count any element twice.
(d) Bonus question: (you don’t need to answer): Is there a geometric way to think about R/ ?
Solution
a)
x-x=0 is in Z for all real numbers x
SO R is reflexive
IF, x-y is in Z then y-x is also in Z
So, R is symmetrix
IF, x-y and y-z are in Z
then x-y+y-z=x-z is also in Z
Hence, R is transitive
Hence, R is an equivalence relation
b)
[1/2]={n+1/2: n is in Z}
c)
Elements of R/~ are of the form:
a+R, where, a is a real number in (0,1]
We need only consider real numbers outside this interval because all other real numbers will
differ by an integer from one of the numbers in this interval.