What is the difference with inequalities?

[Trail_Builder]

hi, I've recently started looking into maths outside of class and am mostly interested in proof.

however, when reading into it a bit i got confused as to what the book meant by a true proof, and a non-proof, and i would like you to clear a few things up for me if you can :D thnx.

this is the non-proof:

Prove that $$\surd2 + \surd6 < \surd15$$

$$\surd2 + \surd6 < \surd15 \Longrightarrow (\surd2 +\surd6)^2 < 15 \Longrightarrow 8 + 2\surd12 < 15 \Longrightarrow 2\surd12 < 7 \Longrightarrow 48 < 49$$

is says the implication is going the wrong way.

so this is the real proof:

Prove that $$\surd2 + \surd6 < \surd15$$

$$\surd2 + \surd6 \geq \surd15 \Longrightarrow (\surd2 +\surd6)^2 \geq 15 \Longrightarrow 8 + 2\surd12 \geq 15 \Longrightarrow 2\surd12 \geq 7 \Longrightarrow 48 \geq 49$$

now i can just about understand how that works. because the implication thing i mean. for example, if M implies N it shows nothing of the truth of N. so, we have to form the negation of $$\surd2 + \surd6 < \surd15$$, which is $$\surd2 + \surd6 \geq \surd15$$, and then prove that there is a contradiction.

What that seems to imply to my nooby brain is that all proofs using inequalities need to be proven by proof by contradiction. Is this right? And then is this the same with equalities?

so does this proof work?

prove $$2^3 + 2*3 = 2*7$$:

$$2^3 + 2*3 = 2*7 \Longrightarrow 8 + 6 = 14 \Longrightarrow 14=14$$

so i havn't proved that one with the equals using proof by contradiction, but then it seems to make sense to me? what is the difference with inequalities? Am I totally confused? Am I totally wrong? can someone please help.

thnx

p.s. excuse my noobishness, i just want to get things right from the get go lol.

 

[matt grime]

There are several things to say.

You're problem is that you're trying to prove it the wrong way. Actually, all the steps in your proof are double implications (if and only if) anyway, so there was no need to explicitly reverse the argument.

The first thing you should stop doing is writing down for proofs an equality or inequality that you want to prove and then manipulating it. I believe what you're doing is called a synthetic proof - start with what you want to show, then work out why its true. The formal proof should then be the reverse steps.

Your second argument with the equality is wrong - the implication is in the wrong direction to be a proof.

 

[Trail_Builder]

hmm, i copied the first nonproof and then the right proof from the book lol. so they must have done a "synthetic proof" lol.

so, are the first proof double implication too or not?

how come the implication is in the wrong direction?

would the first one by right if it started 48 < 49, then showed using the implication things it going to the intial statement?

thnx for the help

 

[matt grime]

You start with something you wish to prove, and then deduce something that is true. This does not show that the original statement is true in the slightest. Shall we think of an example? Does 1=-1? No, but if I square both sides i get 1=1, and that is true...

You *cannot* start from the result you want to show (such as the last 'equality' one) and go from there.

 

[daniel_i_l]

For example, say I want to prove that all cars are red. So I do:
It's a red car => It is a car (since a red car is always a car)
The reason that this proof doesn't work is because we only proved that all red cars are cars, not that all cars are red cars since it only goes one way.

But if we wanted to prove that all red cars have windsheilds then we could write:
It's a red car => It is a car => It has a windsheild

 

[HallsofIvy]

The technique, often called "synthetic proof" and used in proving "identities" in trigonometry is to start from what you want to prove and work back to something that is obviously true. Of course the "real" method of proof should be to start with what is obviously true and work forward to what you wanted to prove. Often we simply make it clear that every step is reversible and leave it to the reader to observe that the "true" proof follows.

 

[Trail_Builder]

oooo i see :D:D:D

thnx for the help guys!

 

[hashimcom]

nice...go forword

dr.hashim