futjim wrote:Hi,

I'm having trouble understanding "the least integer greater than or equal to z" in the following question:

96. For all Z, [z] denotes the least integer greater than or equal to z. Is [x] = 0?

s1) -1 < x < -0.1
s2) [x + 0.5] = 1

What does "the least integer greater than or equal to z" mean? Can someone explain it to me in English? I read this over and over last night and read it again today and I still can't comprehend the meaning to that statement.

Your help is greatly appreciated.

hi..its actaully prety simple..
let us do [2.4]
What are the integers greater than or equal to 2.4?
3,4,5...
What is the least among them?
3
So [2.4] = 3

What is [5]?
Integers greater than or equal to 5 are 5,6,7,...
Leats among them is 5.
so [5] = 5

[] is called a step function. because, the graph looks like steps.

Hope this helps!!

happens to everyone one time or the other

Not sure if this helps much, try replacing "least" with "smallest"

-->"what is the smallest integer that is greater than or equal to z"

Once you see this question, a voice inside your head should say "hmm.. asking for smallest possible integer.. so maybe [z] will be a fraction..hmm... lets see the statements"

Now, if you've got this far then you can go ahead and look at the statements

Statement 1
-1<x<-0.1

--> so x lies between -1 and -0.1 --> the integers greater than x are 0,1,2..... Therefore, [x] i.e. the smallest possible integer greater than x, is 0.
You have a definite value for [x], so statement 1 alone is sufficient

Statement 2
[x + 0.5] = 1

Translating this -> the smallest integer that is greater than or equal to (x+0.5) is the integer 1
Be careful! The question is asking for [x].
Let's try getting different values for [x] and thius prove that this choice is insufficient
(i) assume x = 0.2 --> [0.2+0.5] = [0.7] = 1 ... and [x] = [0.2] = 1
(ii) assume x = -0.3 --> [-0.3+0.5] = [0.2] = 1 ... and [x] = [-0.3] = 0 ... ERROR! ERROR! ERROR! System meltdown


---> [spoiler]Answer is (A) because only statement I alone is sufficient[/spoiler]