The integer part of x is the integer n such that $\displaystyle n\le x < n+1$. (There is exactly one such integer for each real x.) It is usually denoted by [x].
For example:
[3.14]=3
[0.52]=0
[1]=1
[-1.26]=-2
So 2 and 2.5 are in relation, since [2]=2=[2.5], but 1.5 and 2.5 are not.
Hope this helps.
So in this case,
- It IS reflexive since x has the same integer part as x.
- It IS symmetric since if x has the same integer part as y then y has the same integer part as x.
- It IS transitive since if x has the same integer part as y and if y has the same integer part as z then x has the same integer part as z.
correct?
Yes, it is an equivalence relation.
Another way to see this is to notice that you're in fact given a decomposition of R and every decomposition gives you an equivalence relation. (But if you haven't heard much about the correspondence between equivalences and decompositions at your lessons, you should perhaps ignore this comment.)
No.
Take, for instance, the equivalent class of 3.14. It contains all numbers such that x R 3.14.
This is equivalent to
[x]=[3.14]=3.
And what are the numbers such that [x]=3? Precisely the numbers from the interval $\displaystyle \langle 3,4)$, right?
Can you find the remaining classes?