# Homework Help: Is this a necessary and sufficent condition for the floor function?

1. Sep 22, 2006

### mr_coffee

THe book says, state a necessary and sufficent condition for the floor of a real number to equal that number. I looked up what that ment in chapter 1 and it said the following:

r is a necessary and sufficent condition for s means: "r if, and only if, s"

So I scanned through this chapter and i noticed a if and only if statement in the floor description. It says:

Symbolically, if x is a real number and n is an integer, then
[x] = n <=> n =< x < (n+1)

But this isn't saying, when u take the floor of a real number you get that number. Its saying, if you take the floor function of a real number, you get an integer. But the question says, "floor of a real number to equal that number." Or in this case is the real number also an integer?

But the only other thing i found close to what they want is the following sentence but its not an if and only if sentence.

Imagine a real number sitting on a number line. The floor and ceiling of the number are the integers to the immediate left and to the immediate right of the number (unless the number is, itself, an integer, in which case its floor and ceiling both equal the number itself).

I bolded the part that i think might be the necessary and sufficent condition.

So I think they want the following:
The floor of a real number is equal to that number if, and only if, the number is, itself, an integer.

Thanks!

Last edited: Sep 22, 2006
2. Sep 22, 2006

### AKG

I didn't read most of the post, but you're right:

The floor of a real number is equal to that number if, and only if, the number is, itself, an integer.

3. Sep 23, 2006

w00t thanks!