The integer part of x is the integer n such that . (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.)