Integers involving absolute values

hmwin · #1 $Old$ 09-09-2008, 02:28 PM

Assuming a is a non zero integer, which of the following statements is never true...
1. a > -a
2. a < -a
3. [a] = -a
4. [a] = - [a]

any explanation is welcome!

hmwin · #2 $Old$ 09-09-2008, 03:46 PM

So, are you saying that all four answers are false? Could there be more than one false answer?

bkarpuz · #3 $Old$ 09-11-2008, 09:17 AM

Quote:

Originally Posted by Moo $View Post$

Hello,

In general, negative numbers are great counter examples.
For 1. let a be a negative integer (-1,-2,etc... you can choose).
For 2. let a be a positive integer.

False too...

Recall that $|x|=\left\{\begin{array}{ll} -x \quad \text{ if } x<0 \\ x \quad \text{ if } x \ge 0 \end{array} \right.$

Once again, you can tackle it with counterexamples.

Assuming a is a non zero integer, which of the following statements is never true...

1 is true with $a=1$ and 2 is true with $a=-1$ , also 3 is true with $a=-1$ , I think the answer is 4
Because if $a$ is non-zero then $|a|\neq0$ too.

Question. Does $[a]$ denotes the lowest or greatest integer function? If so it is not meaningful since $[a]=a$ holds ( $a$ is an integer).
But it would be nice to consider this question with real $a\neq0$ .
In this case, the solution is 3.

masters · #4 $Old$ 09-11-2008, 11:09 AM

Quote:

Originally Posted by hmwin $View Post$

Assuming a is a non zero integer, which of the following statements is never true...
1. a > -a
2. a < -a
3. [a] = -a
4. [a] = - [a]

any explanation is welcome!

Judging from the title "Integers involving absolute value", I assume that [a] means "the absolute value of a".

So, in #4 [a] = -[a] ==> |a| = - |a|

This would never be true since the absolute value will always produce a positive result.

nioton · #5 $Old$ 10-02-2008, 08:36 AM

Non of the answers are true
if a is not 0 ,it can be positive or nagative ,
check it with one positive for example 3,
then check it with negative for example -3
if these to numbers are not correct for any of the answer,any non ziro integer will act the same

HallsofIvy · #6 $Old$ 10-19-2008, 06:35 AM

Quote:

Originally Posted by nioton $View Post$

Non of the answers are true
if a is not 0 ,it can be positive or nagative ,
check it with one positive for example 3,
then check it with negative for example -3
if these to numbers are not correct for any of the answer,any non ziro integer will act the same

The question asked was "which of the answers are NEVER true".

The first, a> -a, is true whenever a is positive and so is not "never true".

The second, a< -a, is true whenever a is negative.

The third and fourth, |a|= -a and |a|= -|a| are true for a= 0.

NONE of the statments is "NEVER true".