Quote: Original post by nullsquared
Hm, I see what you're saying. The only problem now is that we have two contradicting views in the thread [grin]; I hope a few more people come in to confirm one way or the other.

Do you care to show what you believe to be the opposing viewpoint? Sneftel is clearly correct, but it's possible that you're misinterpreting what he's saying.

Quote: Original post by cowsarenotevil

Quote: Original post by nullsquared
Hm, I see what you're saying. The only problem now is that we have two contradicting views in the thread [grin]; I hope a few more people come in to confirm one way or the other.

Do you care to show what you believe to be the opposing viewpoint? Sneftel is clearly correct, but it's possible that you're misinterpreting what he's saying.

This:

Quote: Original post by aryx
Right, you definitely don't start by assuming what you have is true. That makes for quite the meaningless argument. What you've shown is fine if reversed, but to start your proof with what you're trying to prove would mean being marked wrong by anyone in the [pure] math business. This is how I would perhaps write things:

1 = H^2 / H^2  = (O^2 + A^2) / H^2 // expand H^2 based by Pythagorean theorem  = O^2 / H^2 + A^2 / H^2 // distribute  = (O/H)^2 + (A/H)^2 // rearrange  = sin^2 θ + cos^2 θ // substitute

This clearly shows how you manipulated one side to obtain the opposite side. You began with something that was true (1 = H^2 / H^2) and manipulated it to draw a conclusion.

vs. this:

Quote: Original post by Sneftel
That's a perfectly acceptable proof, because it's exactly the same thing; it's just written the other way around. Just as in my formulation each statement is supported by the statement before it, in this formulation each statement is supported by the statement after it. In my formulation each statement is provably true as soon as it's written down; in this formulation nothing is proven until you get to the last line, at which point justification ripples back through the proof to the beginning. So because it's written this way, each line has to lead logically to the previous line. If you put in 0=0 somewhere, that would no longer be a valid proof, because the line before it would not be adequately justified. But as long as we don't change the left hand side, we're guaranteed that statement n is justified by statement n+1, due to transitive equality. In fact we could change the right hand side and still have this property, as long as we didn't change both sides at the same time.

I don't think we disagree, except on notation. I've often seen the "only change one side" form, but it's limited to proving equations, so it's pretty uncommon among proofs as a whole. In either case, the sufficiency of the proof is clearly verifiable from the statements given.

This is an issue which has always bothered me (precal HS, precal Uni, precal tutoring friends/family most recently). Scouring the web led me to contradictory and questionable answers for hours until I found a thread on another forum on the philosophy of trig identities. There were a number of claims in this thread (not the linked one) that I did not like, but the given link addresses them as well. The OP's thread sums up my problems, and the responses, unless shown to be incorrect, I have found to be satisfactory.

Quote: Original post by Sneftel
I don't think we disagree, except on notation. I've often seen the "only change one side" form, but it's limited to proving equations, so it's pretty uncommon among proofs as a whole. In either case, the sufficiency of the proof is clearly verifiable from the statements given.

I've also seen both, but like I said I've seen math teachers use implication arrows if there's any doubt about which way the implication goes.

To be honest, though, a person would have to be kind of dumb to see a proof that starts with the statement that is supposed to be proven and not at least consider the possibility that it is, in fact, a correct proof written "backwards."

Quote: Original post by Sneftel
I don't think we disagree, except on notation. I've often seen the "only change one side" form, but it's limited to proving equations, so it's pretty uncommon among proofs as a whole. In either case, the sufficiency of the proof is clearly verifiable from the statements given.

Actually, we do disagree to a certain extent. By starting with sin^2 + cos^2 = 1 and leading up to a trivially true statement (sin^2 + cos^2 = sin^2 + cos^2), you haven't really proved anything. In particular, what you've shown is that by assuming sin^2 + cos^2 = 1 is true, you can show that sin^2 + cos^2 = sin^2 + cos^2. This is why one should reverse the logic; start with something that is true, and draw conclusions from that.

Just to rewrite it to show what I mean. Written this way:

sin^2 θ + cos^2 θ = 1sin^2 θ + cos^2 θ = H^2 / H^2 // multiply by the equivalent of 1/1sin^2 θ + cos^2 θ = (O^2 + A^2) / H^2 // expand H^2 based by Pythagorean theoremsin^2 θ + cos^2 θ = O^2 / H^2 + A^2 / H^2 // distributesin^2 θ + cos^2 θ = (O/H)^2 + (A/H)^2 // rearrangesin^2 θ + cos^2 θ = sin^2 θ + cos^2 θ // substitute

would be regarded as this:

sin^2 θ + cos^2 θ = 1=> sin^2 θ + cos^2 θ = H^2 / H^2 // multiply by the equivalent of 1/1=> sin^2 θ + cos^2 θ = (O^2 + A^2) / H^2 // expand H^2 based by Pythagorean theorem=> sin^2 θ + cos^2 θ = O^2 / H^2 + A^2 / H^2 // distribute=> sin^2 θ + cos^2 θ = (O/H)^2 + (A/H)^2 // rearrange=> sin^2 θ + cos^2 θ = sin^2 θ + cos^2 θ // substitute

unless you read from bottom to top of course (=> is implication). From this you can easily see that you haven't proven anything but the fact that by assuming that sin^2 + cos^2 = 1 is true, you can show that sin^2 + cos^2 equals itself. If those were equivalencies, you'd be safer, but if it is not explicitly shown then implications are what are assumed (that one step leads to the next, not the other way around).

It's clear to most how the proof should go, but it's best to be explicit, and never to start with the conclusion. I'm just trying to lead nullsquared down a more elegant and easy-to-follow proofing path. I know that my first real analysis and discrete math courses were a little challenging until I got the logic down right. I didn't do that great in my first real analysis course, but several real analysis courses later I was acing them :)

Quote: Original post by cowsarenotevil
To be honest, though, a person would have to be kind of dumb to see a proof that starts with the statement that is supposed to be proven and not at least consider the possibility that it is, in fact, a correct proof written "backwards."

The question is: does that make it a valid proof? I can write this sentence backwards, but does that make it a valid sentence? After all, it has all the same words and punctuation.

Quote: Original post by cowsarenotevil

Quote: Original post by Sneftel
I don't think we disagree, except on notation. I've often seen the "only change one side" form, but it's limited to proving equations, so it's pretty uncommon among proofs as a whole. In either case, the sufficiency of the proof is clearly verifiable from the statements given.

I've also seen both, but like I said I've seen math teachers use implication arrows if there's any doubt about which way the implication goes.

To be honest, though, a person would have to be kind of dumb to see a proof that starts with the statement that is supposed to be proven and not at least consider the possibility that it is, in fact, a correct proof written "backwards."

True enough, but best to make it clear how the logic flows. It could make the difference between full and partial marks on a pure math exam ;)

Quote: Original post by aryx
True enough, but best to make it clear how the logic flows. It could make the difference between full and partial marks on a pure math exam ;)

Well, yeah, but I've also run into plenty of cases where the proof only becomes obvious if I start it "backwards" in which case doing it forward would have resulted in no credit at all. Of course, I always take the ten seconds required to draw backwards arrows on each line in this case.

Quote: Original post by aryx
Actually, we do disagree to a certain extent. By starting with sin^2 + cos^2 = 1 and leading up to a trivially true statement (sin^2 + cos^2 = sin^2 + cos^2), you haven't really proved anything.

Sure you have. Given E=E, you've shown that since E=D, D=C, C=B, and B=A, E=A. I guess the notation's a bit confusing if you haven't seen it and are used to two-column proofs, but it's a convenient form for constructing identities.