Direct Proof: |x|=0 Implies x=0 - Shorter Proof Available?

[evagelos]

In proving :|x|=0$$\Longrightarrow$$ x=0,the following indirect proof is offered:

Let |x|= 0 and suppose $$x\neq 0$$ then x>0 or x<0.

For x>0 |x| =x and since |x|=0 ,x=0

For x<0 |x| = -x and since |x|=0 ,x=0

So in either case x=0 ,which is a contradiction since we assumed $$x\neq 0$$

Hence x=0.

Is there a direct shorter proof??

 

[Stephen Tashi]

If you consider three cases instead of two:
case 1) x= 0
case 2) x < 0
case 3) x > 0
then you get a direct proof that's not much different.

 

[evagelos]

If you consider three cases instead of two:
case 1) x= 0
case 2) x < 0
case 3) x > 0
then you get a direct proof that's not much different.

How ?? please show me

 

[Stephen Tashi]

Case 1) x = 0. Then |x| = |0| = 0 = x

The other two cases are handled in the proof you wrote

 

[evagelos]

Case 1) x = 0. Then |x| = |0| = 0 = x

The other two cases are handled in the proof you wrote

We do not want to prove: if x=0 ,then |x|=0,

But : if |x|=0,then x=0

So you cannot start with x=0

 

[Stephen Tashi]

We do not want to prove: if x=0 ,then |x|=0,

But : if |x|=0,then x=0

So your cannot start with x=0

You can assume x= 0, if you cover all cases and x = 0 is the case that you are considering.

The hypothesis of the theorem is: |x| = 0
The conclusion is: x = 0

There are 3 possible cases for x. They are: x = 0 , x > 0, x < 0


Case 1: Assume x = 0. We determine |x| satisfies the the hypothesis and that x satisfies the conclusion.

Case 2: Assume x > 0. We determine |x| does not satisfy the hypothesis so the value of x is irrelevant to validity of the theorem.

Case 3: Assume x < 0. We determine |x| does not satisfy the hypothesis, so the value of x is irrelevant to validity of the theorem.

 

[evagelos]

You can assume x= 0, if you cover all cases and x = 0 is the case that you are considering.

The hypothesis of the theorem is: |x| = 0
The conclusion is: x = 0

There are 3 possible cases for x. They are: x = 0 , x > 0, x < 0


Case 1: Assume x = 0. We determine |x| satisfies the the hypothesis and that x satisfies the conclusion.

Case 2: Assume x > 0. We determine |x| does not satisfy the hypothesis so the value of x is irrelevant to validity of the theorem.

Case 3: Assume x < 0. We determine |x| does not satisfy the hypothesis, so the value of x is irrelevant to validity of the theorem.

I suppose we have to go back to the basics :

What is a mathematical proof.

So as to find out if what you have written is a proof

 

[Stephen Tashi]

What I've written is not a mathematical proof unless you fill-in the details using methods in your first post. (For example: Case 2 needs the work: Assume x > 0,. Then |x| = x > 0 So |x| > 0. Thus |x| does not satisfy the hypothesis.)

In practical terms, a proof is any argument that will convince the population of experts that you wish to convince.

If you get into specialized fields of logic (for example, "mechanical theorem proving" by using algorithms) then the rules for what constitute a proof become more strict.

 

[evagelos]

What I've written is not a mathematical proof unless you fill-in the details using methods in your first post. (For example: Case 2 needs the work: Assume x > 0,. Then |x| = x > 0 So |x| > 0. Thus |x| does not satisfy the hypothesis.)

In practical terms, a proof is any argument that will convince the population of experts that you wish to convince.

If you get into specialized fields of logic (for example, "mechanical theorem proving" by using algorithms) then the rules for what constitute a proof become more strict.

So if some of the experts are convinced and some are not ,what happens then?

In our case now what must we do to exactly know whether what you have written constitutes a proof or not

 

[chiro]

So if some of the experts are convinced and some are not ,what happens then?

That is where it gets interesting.

Both sides enter into debate and hopefully both parties are open to the other sides argument and hears the other side out before presenting their argument.

But you have to remember human beings are emotional, arrogant, egotistical and so on. Also I think you can imagine that if someone has learned something and has spent pretty much more than half their life learning it, its going to be very interesting to see how that side takes the argument.

It's like say telling a nun that God doesn't exist and getting into a debate. The nun is probably going to be a little annoyed.

Having said that, if that stuff is out of the way, it is usually a good exercise for not only both sides in the debate, but also any spectators that are watching on in interest.

This is, in short, the real world.

 

[evagelos]

That is where it gets interesting.

Both sides enter into debate and hopefully both parties are open to the other sides argument and hears the other side out before presenting their argument.

But you have to remember human beings are emotional, arrogant, egotistical and so on. Also I think you can imagine that if someone has learned something and has spent pretty much more than half their life learning it, its going to be very interesting to see how that side takes the argument.

It's like say telling a nun that God doesn't exist and getting into a debate. The nun is probably going to be a little annoyed.

Having said that, if that stuff is out of the way, it is usually a good exercise for not only both sides in the debate, but also any spectators that are watching on in interest.

This is, in short, the real world.

Bottom line we will never find out whether the above proof is right or wrong

And how about the experts in this forum ,do they approve of this proof??

 

[chiro]

Bottom we will never find out whether your proof is right or wrong

And how about the experts in this forum ,do they approve of your proof??

I think this is the great part about science and also about life: the fact that things aren't clear cut, that require a lot of thought, effort, discussion, debate, and development.

If life was so clear cut where would the adventure or learning be? If you were never surprised, what would that do to your life? Being surprised can be bad, but it can also spice things up.

The truth is most of life is like this. Most things in the world aren't black or white: they are black and white and are hard to reduce absolutely.

Personally I'm glad that, at least for a lot of us, we've gone from burning witches at the stake to a jury based trial and forums that are open to almost anyone to make a contribution that is judged (in most circumstances) on the content of the message over the messenger itself: it's great!

 

[Stephen Tashi]

evangelos,

As chiro points out, if a proof fails to convince an actual expert, the expert will have specific objections to it that can either be answered or admitted to be defects.

In addition to objecting to the method of a proof, an expert may say that thing we attempted to prove is false. He can try to give a counterexample to show this.

On the forum, there are experts whose opinions I respect and then there are those who I wouldn't bother trying to convince of anything. For example, if you look in threads on topics like "Is .9999... really 1", "What is multiplication?" and so forth, you find posts by experts and also by a variety of people with strong opinions but no competence in rigorous math.

For a person taking a course or exam, the only experts that matter are the instructor and the grader!

 

[evagelos]

I think this is the great part about science and also about life: the fact that things aren't clear cut, that require a lot of thought, effort, discussion, debate, and development.

If life was so clear cut where would the adventure or learning be? If you were never surprised, what would that do to your life? Being surprised can be bad, but it can also spice things up.

The truth is most of life is like this. Most things in the world aren't black or white: they are black and white and are hard to reduce absolutely.

Personally I'm glad that, at least for a lot of us, we've gone from burning witches at the stake to a jury based trial and forums that are open to almost anyone to make a contribution that is judged (in most circumstances) on the content of the message over the messenger itself: it's great!

Is the proof given by Stephen Tashi according to your opinion correct??

 

[daveb]

And how about the experts in this forum ,do they approve of this proof??

Other than there is one small mistake for the case x<0. It should read |x| = -x, but since |x| = 0, then -x=0 which implies x = 0, but otherwise it's good.