Anyone good at MATH PROBLEMS?

Question

i need help with my geometry homework.... can u guys atleast answer one of these questions.
DIRECTIONS: SHOW CONJECTURE IS FALSE BY FINDING A COUNTEREXAMPLE.

1.) THE DIFFERENCE OF TWO WHOLE NUMBERS IS A WHIOLE NUMBER.

2.) THE ABSOLUTE VALUE OF THE SUM OF TWO NUMBERS IS THE SUM OF THEIR ABSOLUTE VALUES, MEANING...Ia+bI = IaI + lbl.

3.) IF m doesnt equal -1, then m/m-1 > 1.

Answer 1

For number 1

Take any whole number and subtract it by a larger whole number:

1 - 2 = -1, and since -1 is not a whole number you have your counterexample.


For Number 2,
Simply take a number and it's opposite:

|1 + -1| = 0
|1| + |-1| = 2

Another counterexample.


For Number 3,
Take 0:

m = 0
0 / 0 -1 = 0/-1 = 0
And since 0 < 1, you have your third counterexample.

Answer 2

1. The set of whole numbers includes the set of natural (counting) numbers as well as zero. The set of whole numbers does NOT include negative integers. Hence, any example of a - b where a < b will be a counterexample because the result will be negative. For example,

3 - 4 = -1, which is not a whole number because it is negative.

2. The assertion will be false if a and b have opposite signs. Counterexample would be a = -5, b = 2.

|a+b| =? |a| + |b|
|-5 + 2| =? |-5| + |2|
|-3| =? 5 + 2
3 ≠ 7 so the assertion is false.

3. I assume that this is supposed to read
"If m ≠ 1, then m / (m-1) > 1."
which will be false for any value of m < 1, such as m = 0:
m / (m-1)
= 0 / (0 - 1)
= 0 / (-1)
= 0, which is definitely not greater than 1.

Answer 3

1) Here is a counter example. 3 - 5 = -2.
3 is a whole number and 5 is a whole number. Their difference is not a whole number. In general for any whole number x and y. If x < y, then x - y is not a whole number.

2) |a + b| = |a| + |b|. But if a > 0 and b < 0 and |a| <=|b| ,
|a+b| = -(a+b). |a| = a and |b| = -b

therefore, -(a+b) ╪ a - b.

With number, you will see it better. Let a = 2 and let b = -3
we will have |a + b| = | 2 -3 | = |-1| = 1
But |a| + |b| = |2| + | -3| = 2 + 3 = 5. Hence 1 ╪ 5. That is a counter example

3) I think you meant m/(m-1) >1. That is m-1 is in the denominator.
In case you meant that.

let m = ½ , then ½ / (½ -1) = -1 and it is less than 1.

Answer 4

I think you'll have trouble finding a counterexample to (1) because it's true. The integers form a group over addition and are therefore closed over addition and, by corollary, subtraction.

(2) is easy, just make one of the numbers negative and the other positive.

(3) you have me stumped. I can't find a counterexample. maybe it's true?

Edit: oops, yes of course, if m<1, then (3) is easy to solve.

Answer 5

1) this conjecture is false if zero is not included in the definition, in which case 1 - 1 = 0
2) |1 + -1| = 0 and |1| + |-1| = 2
3) 1/1 - 1 > 1 ==> 0 > 1 = false

Answer 6

1) I can't think of any counterexamples to this, unless by "whole numbers" they mean positive integers and not just integers.

2) One counterexample is a= -1 and b= 1

3) Try m = 1/2

Answer 7

1). 2-4 = -2, which is not a whole number.
2). Let a = 2 and b = -2.
Then |a+b| = 0, but |a| + |b| = 4.
3). Let m be negative, e.g., m = -2
Then m/(m-1) = -2/-3 = 2/3 < 1.

Answer 8

1.) Got me stumped. And for the record, negative integers are still whole numbers.

2.) False if one of the numbers is negative. For example:

|5+(-3)| = |2| = 2
Whereas:
|5| + |-3| = 5 + 3 = 8

3.) If I understand how it is written, False for any negative m. Also false for zero. Hmm...maybe false for any m < 1.

Answer 9

1) Counter example: 2-3 = -1, -1 is not a whole number.
2) Counter example: |2-3| = 1 while |2|+|-3| = 5
3) Counter example: m = 0.5, then 0.5/(0.5-1) = -1 < 1
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Ideas: If you can find a counter example, then the conjecture is false.
It seems that a lot of guys don't remember the definition of a whole number:0, 1, 2...n, ...

Answer 10

1. I'm pretty sure #1 is true...

2. This is false: use a=2 and b = -3. |2-3| = 1 where |2| + |-3| = 5.

3. This is false: use m = 0. 0/0-1 = -1 which isn't greater than 1.