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# If [z] denotes the greatest integer less than or equal to z and [z] =

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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
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ammuseeru wrote:
Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, [2]=2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

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Hope it helps.

If question asks " [z] denotes the greatest integer less than or equal to z and [z] = 0", then can we say 0<=Z<1 ?

Regards,
Ammu

Yes. [z] = 0, would mean that $$0\leq{z}<1$$ --> ANY number from this range when rounded down to the nearest integer is 0.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
1
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Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, [2]=2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

Similar questions to practice:
if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html
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for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html
is-x-an-odd-integer-1-x-2-2x-99856.html
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is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html

Hope it helps.

Hi Bunuel,

Just a question. Why do you think it could not be A?
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
3
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gbascurs wrote:
Bunuel wrote:
sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 <= z < -1
C. -2 < z <= -1
D. -1 <= z < 0
E. -1 < z <= 0

[z] is the greatest integer less than or equal to z means that some function [] rounds DOWN a number to the nearest integer. For example [1.5]=1, [2]=2, [-1.5]=-2, ...

Now, since [z] = -1, then $$-1\leq{z}<0$$ --> ANY number from this range when rounded down to the nearest integer is -1.

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for-all-numbers-t-let-t-be-defined-as-the-greatest-105416.html
is-x-an-odd-integer-1-x-2-2x-99856.html
for-all-z-denotes-the-least-integer-greater-than-or-110561.html
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if-is-the-greatest-integer-less-than-or-equal-to-x-what-144277.html
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is-the-greatest-integer-less-than-or-equal-to-the-real-n-154006.html

Hope it helps.

Hi Bunuel,

Just a question. Why do you think it could not be A?

Because z is not necessarily -1, it could be any number from -1 to 0 ($$-1\leq{z}<0$$ ), for example, -1/2.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
But the question states that it is an integer... What do I missing?
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
gbascurs wrote:
But the question states that it is an integer... What do I missing?

The questions says that [z] is an integer, not z.

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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
Bunuel wrote:
gbascurs wrote:
But the question states that it is an integer... What do I missing?

The questions says that [z] is an integer, not z.

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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
Dear Bunuel,

Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given.
Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam.
What trick am I missing here?

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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
Dear Bunuel,

Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given.
Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam.
What trick am I missing here?

The question asks which of the following statements MUST be true, not could be true. z could be -1, but it could also be, for example, -1/2. Thus A is not a statement which must be true.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
Bunuel wrote:
Dear Bunuel,

Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given.
Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam.
What trick am I missing here?

The question asks which of the following statements MUST be true, not could be true. z could be -1, but it could also be, for example, -1/2. Thus A is not a statement which must be true.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.

I really cannot understand why A cannot be the answer.

Z=-1

[Z] = -1

Condition satisfies.

I have read the theory and solved all the questions in the link, I don't know what am I missing
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
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anewbeginning wrote:
Bunuel wrote:
Dear Bunuel,

Although I completely agree that the answer is indeed option D here, however option A is not wrong also. If we plug in the value given in option A, we indeed satisfy the condition given.
Does the inclusion of option A not make choosing the correct answer slightly ambiguous. I mean why give an option such as A at all in the exam.
What trick am I missing here?

The question asks which of the following statements MUST be true, not could be true. z could be -1, but it could also be, for example, -1/2. Thus A is not a statement which must be true.

Check other Rounding Functions Questions in our Special Questions Directory.

Hope it helps.

I really cannot understand why A cannot be the answer.

Z=-1

[Z] = -1

Condition satisfies.

I have read the theory and solved all the questions in the link, I don't know what am I missing

The question is a MUST BE TRUE for ALL values. Not a could be true. Had this question been a 'could be true" question, then you would have stopped at option A itself and moved to the next question.

But as this is a MUST BE TRUE question, you need to make sure that [z] =-1 is ONLY satisfied by z =-1 from the given options (and not by any other option). The correct answer to a MUST BE TRUE question, will nullify all other options. This is a very important point to remember. We are not denying that z = -1 is not true but what we are mentioning is that is it the ONLY possible choice ? Not necessarily z = -0.5 also satisfies the given condition.

So if z = -0.5 also satisfies the given condition, then option D is true as well (along with option A, per your statement). A question can not have 2 correct answers.

As I said before, in a MUST BE TRUE question,

1. Get the option(s) that satisfy a given condition
2. Eliminate all but 1 options. The remaining option will be the correct answer. There will always be concrete reasons to eliminate other options. If you are not able to eliminate options, then you are missing some important information.

Unless you take care of both the options, you can not be sure of your answer.

For this question, I adopted the following method:

1. Used z =-0.5 to eliminate options A-C
2. Used z = 0 to eliminate option E. If z =0 then [z] = 0 and $$\neq$$ -1

The only option remaining after the 2 steps above was D and is the correct answer.
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
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sguptashared wrote:
If [z] denotes the greatest integer less than or equal to z and [z] = -1 which of the following statements must be true?

A. z = -1
B. -2 ≤ z < -1
C. -2 < z ≤ -1
D. -1 ≤ z < 0
E. -1 < z ≤ 0

Attachment:
prep.JPG

Let's make sure we understand what this [] symbol means.
Here are a few examples:
[3.1] = 3
[5.8] = 5
[0.7] = 0
[-0.9] = -1
[-4.6] = -5

So, if [z] = -1, then -1 ≤ z < 0

Cheers,
Brent
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Re: If [z] denotes the greatest integer less than or equal to z and [z] = [#permalink]
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