Assume real numbers R for now Consider relation on R x y iff

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.