Conditional vs Equivalent Statements

[Mates]

When I was doing calculus I and II, my professor didn't really explain clearly how exactly the arrows work when making "chains" of equations to prove some question/assumption.

I know that statement (A) If I have an apple --> I have a fruit. must go in one direction. But I don't understand why when it comes to using equations to prove something.

For example, he once docked me marks for putting the arrows in the wrong direction when proving some trig problem. To this day I still don't know what the difference is when it comes to using arrows with equations. Shouldn't proofs with only equations always be IFF because they should always be individually true and not contingent on the former or latter equation?

Can someone give me an example where a simple proof can go one direction but not the other?

 

[pka]

When I was doing calculus I and II, my professor didn't really explain clearly how exactly the arrows work when making "chains" of equations to prove some question/assumption. I know that statement (A) If I have an apple --> I have a fruit. must go in one direction. But I don't understand why when it comes to using equations to prove something.

For example, he once docked me marks for putting the arrows in the wrong direction when proving some trig problem. To this day I still don't know what the difference is when it comes to using arrows with equations. Shouldn't proofs with only equations always be IFF because they should always be individually true and not contingent on the former or latter equation?

First, I simply have no idea what exactly you are asking. It seems as though you have mixed up some ideas employed in the study of logic with methods employed in proof theory. I have never thought about using arrows in mathematical proofs involving equations. It is true that every proof must conform to proper logical deductions. It is also true that a large number of logical deductions are dependent upon inference (if - then) statements: \(\displaystyle P \Rightarrow Q\) is read "if P then Q". As a matter of short hand one might write \(\displaystyle D \Rightarrow C\) for "differentiability implies continuity. However, one should never do so in a formal proof.

You may want to further clarify and expand your examples?

 

[Mates]

I only remember that the question asked us to prove that A = B, where A and B had trig functions.

I couldn't remember the correct identities needed to "turn the left side into the right side", so I tried to simplify both sides simultaneously. What I did to one side, I did to the other until I came to something clear like 2 = 2.

So let's just use, for example, sin0 = (1- (cos0)^2)^(1/2). Just imagine it was much trickier.

Instead of using identities to make one side look like the other, I applied a series of operations to both sides until I got something obvious like tan0 = tan0.

Let's assume that I started with the question sin0 = (1- (cos0)^2)^(1/2) then the arrow would point to something like ------> (sin0)^2 = 1 - cos0, then -----> ... until I had something obviously true.

He said that it would have actually been okay if the arrows had gone the other way, starting from the trivial end. He also said that I should never get into the habit of doing "proofs" this way because sometimes the arrow won't be true no matter which way it goes even for proofs like these.

I don't understand what he was talking about. I don't even understand the purpose of the arrows.
in general, do it the way I did it.

 

[Mates]

I am not actually disagreeing with docking me marks; I just wanted a clear understanding on when to use arrows. He would use arrows in situations for simplifying, like "show x = 6" from, say, 2x + 3 = 15.

Is this right: 2x + 3 = 15 ---> 2x = 12 ---> x = 6?

 

[HallsofIvy]

I am not actually disagreeing with docking me marks; I just wanted a clear understanding on when to use arrows. He would use arrows in situations for simplifying, like "show x = 6" from, say, 2x + 3 = 15.

Is this right: 2x + 3 = 15 ---> 2x = 12 ---> x = 6?

Yes, you are saying that "from 2x+ 3= 15, it follows that 2x= 12, and, from that, that x= 6.

 

[Mates]

Yes, you are saying that "from 2x+ 3= 15, it follows that 2x= 12, and, from that, that x= 6.

Could I have put the arrows in the other direction (with the same positions of the equations but with the arrows pointing to the left)?

 

[ksdhart2]

Could I have put the arrows in the other direction (with the same positions of the equations but with the arrows pointing to the left)?

I guess you could...

\(\displaystyle x=6 \implies 2x=12 \implies 2x+3=15\)

That's logically consistent. "If x equals 6, then 2 times x equals 12, then 2 times x plus three equals 15." However, in general, I'd caution against doing that because it often involves assuming the very thing you're trying to prove.

 

[Mates]

Okay, I think I am clear. Thanks everyone

 

[stapel]

I only remember that the question asked us to prove that A = B, where A and B had trig functions.

I couldn't remember the correct identities needed to "turn the left side into the right side", so I tried to simplify both sides simultaneously. What I did to one side, I did to the other until I came to something clear like 2 = 2.

Just to be clear, what you did, described above, is not, in technical terms, a proper proof. But it is the "magic" that goes into those proofs from which you have no idea the source of the steps.

The author of a "magical" proof did the work you did, but did it on scratch paper. Then he started on one side, worked down to that "2 = 2" line, and then worked backwards, going "up" the other side until he arrived at the other side of the original equation.