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I need to show that A = B. Do I have to only work with one side of the equation and show that it equals the other side without using the other side? Or can I assume that they equal each other and use both sides together to show equality?

For example, if I had to show that $x^2/x = x$, do I need to either simplify the left side by making it $x$ or multiply the right side by $\frac{x}{x}$, or am I allowed to set the equations equal to each other and multiply $x$ by both sides, showing that $x^2 = x^2$ ?

It doesn't seems like it is supposed to be a proof, since it doesn't use the word "prove," but I am unsure.

Nate
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  • "Show " means the exact same thing as "Prove ". – xavierm02 Nov 11 '13 at 14:01
  • you can start your way from any side reaching the other which ever side you find to be more easy – Deiknymi Nov 11 '13 at 14:01
  • It really depends what A and B actually ARE! Are they numbers, polynomials, vectors, matrices, sets, groups, rings, fields, modules, vector spaces, categories...? – fretty Nov 11 '13 at 14:03
  • The one thing you should not do is to start by putting $A=B$, that would indicate you are assuming the thing you are trying to prove. – Old John Nov 11 '13 at 14:05
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    Working with regular algebra, be careful if you want to try the "do it to both sides" approach. Only operations that are one-to-one can be used if you want to be certain that it's correct. For instance, if you start with $-1=1$, you can square both sides to get $1=1$, but of course the starting equation was false, despite the result being equal. – Glen O Nov 11 '13 at 14:07
  • Okay, this is what I am trying to show, or prove:

    (z^n-w^n)/(z-w) - nw^(n-1) = (z-w)(sum(k=1,n-1) kw^(k-1)*z^(n-k-1)

    And what I have reduced the right side to is:

    (z-w)[((n-1)(w^(n-1))/(w-z) - ((z)(w^(n-1)-z^(n-1))/(w-z)^2)]

    So I guess I can just keep simplifying that to equal the left side and I am good?

     – Nate Nov 11 '13 at 14:15
  • I didn't check if what you have is correct (you could try asking in a different question), but if it is, and you can simplify it to the left side, then yes, that's an acceptable proof. Don't forget Michael Hoppe's answer though and make sure you assume $z\neq w$. – Casteels Nov 11 '13 at 15:06

4 Answers4

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Since you don't yet know that the two sides are equal, you should, in a proof, start from one side and end at the other.

That being said, to find hints on how to do that, a common strategy is to assume the equation is correct and look at simplifying steps, but presenting this as the actual solution is circular logic. Sometimes the proof is simply a matter of reversing those steps!

Casteels
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When your instructor says "show," s/he really means "prove!"

The logic behind the proofs that work on one side or the other side or both sides is really the same in each case. if you have that A=B, and you also have that B=C, then it is fair game in a proof to conclude that A=C. If, for example, you wanted to work with A, you could first show that A=C, and then that C=D, and then that D=B. That is, you could show that A=C and C=D and D=B. For then it would be fair game to conclude that A=B. If you started from B, you could show that B=X and X=Y and Y=A. From which you could conclude that B=A. But you could also work from both sides. You could show that A=W and W=Z, and then you could show that B=V and V=Z. And then you would have A=W W=Z and Z=V and V=B, so that, as usual, it would be fair to conclude that A=B.

dan
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Sometimes you start from $A=B$ then deduce something true. But after doing that, reverse your steps for the final write-up, so that is starts with something true, and concludes $A=B$. And (of course) check that your steps are correct when done backward.

GEdgar
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There is no one technique; you can use any of the following, depending on what's easiest for your particular problem:

1) Start with $A$, and perform manipulations to turn it into $B$.

2) Start with $B$, and perform manipulations to turn it into $A$.

3) On a piece of scratch paper, start with $A=B$ and use manipulations to turn it into an equation where both sides are equal, $C=C$. This is the scaffolding, not the proof! Now in your formal proof, start from $C=C$, and use the reverse of the manipulations you worked out to turn it into $A=B$.

Sometimes people, as a shortcut, write proofs that start with $A=B$, and derive $C=C$, and claim that this a valid proof that $A=B$ because "all steps are reversible." Often this is correct, but it's like not wearing a seatbelt because you know you're only driving to the drugstore two blocks away: it's lazy and not strictly correct, probably won't get you killed once you know what you're doing, but also not something to be encouraged when you're first starting out.

user7530
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