partition for each equivalence relation

hello, i am trying to figure out what i need to do for this problem

for (x,y) is an element of reals, xRy iff x-y is an element of integers

i am supposed to describe the partition for the equivalence relation. im not exactly sure how to do this. i have a similar example, which i don't really understand either...this example is

for n,m is an element of integers, nRm iff n+m is even

now this answer would be {X,Y} where X is the set of even intergers and Y is the set of odd integers.

i don't really understand why this would be so, because odd + even= odd

coudl someone please help me understand what is going on here? thank you