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Abstract
Chapter 1 surveys some common forms of mathematical proofs.
My experience is that many students are uncomfortable with this, but we are going to be seeing a lot of proofs in this course and you will be responsible for writing a number of them.
Here I discuss some common mistakes and problems that I have encountered in student-written proofs.
The various forms of proof discussed in Chapter 1 of the text are not given in great detail. That’s because these are not supposed to be new. If you are really struggling with them, you may need to do some review quickly. But don’t expect to find that in advanced texts or resources specific to automata theory and languages. Look for much more basic mathematics resources instead.
For example, proof by induction, something that students seem to struggle with a lot, is according to the Virginia Standards of Learning, a pre-algebra (early high-school) concept. So, yes, you should be comfortable with the concept by now. No, it’s not covered in depth in the textbook, because that’s only intended as a refresher.
In general a proof is an argument to convince the reader of a conclusion. Its purpose is to convince the reader. It’s not the reader’s responsibility to sift through the argument for pieces that can be re-arranged to make sense.
I often see student-written proofs that consist of line after line of mathematical equalities, with no obvious relation from one to the next (and no English words to explain how they were supposed to be related).
In many of these cases, I truly believed that they were simply unrelated lists of “facts” (e.g., formulas copied from the textbook), after which appeared a simple assertion that, altogether, these proved the theorem. In other cases, I suspected that there was some coherent idea in the student’s mind, but could only guess at what it might be.
That doesn’t work. You have to
That communication generally requires adding text between the steps of mathematics.
I may often ask for a proof using a specific technique.
In that case, I expect you to give me a proof using that method, even if the theorem is already proven in the textbook using a different style of proof.
Many proofs involve showing two expressions to be equal. You’ve been doing that since you first studied algebra, but there are some subtleties to chaining together a series of steps that actually make sense.
You can’t prove that $\mbox{expression}_1$ equals $\mbox{expression}_2$ by writing
\[ \begin{align} \mbox{expression}_1 = & \mbox{expression}_2 \\ \mbox{expression}_{1a} = & \mbox{expression}_{2a} \\ \mbox{expression}_{1b} = & \mbox{expression}_{2b} \\ \vdots & \\ \mbox{expression}_{n} = & \mbox{expression}_{n} \\ \end{align} \]
in which the a, b, … variants are applied by applying the same operation to both the left-hand and right-hand sides of the preceding line, until the left-hand and right-hand sides have been reduced to something identical on both sides.
A proof like this results in a series of perfectly correct steps that add up to absolutely nothing.
This does not prove that $\mbox{expression}_1$ equals $\mbox{expression}_2$. Instead, it proves that if $\mbox{expression}_1$ equals $\mbox{expression}_2$, then $\mbox{expression}_n$ equals $\mbox{expression}_n$. But that’s kind of useless, since we already know that any expression equals itself.
For example, suppose that I wished to show, as part of a proof by induction, that, if $\sum_{i=0}^{k} 1 = k + 1$ then it must be true that $\left(\sum_{i=0}^{k+1} 1\right) = k + 2$.
Here is an invalid proof in a form that I see all too often:
Prove that
\[ \left(\sum_{i=0}^{k+1} 1\right) = k + 2 \]
Expand the sum:
\[ \left(\sum_{i=0}^{k} 1\right) + 1 = k + 2 \]
We have stipulated that $\sum_{i=0}^{k} 1 = k + 1$, so substituting
\[ \left( k + 1 \right) + 1 = k + 2 \]
Subtract $k+1$ from both sides.
\[ 1 = 1 \]
which is, obviously true.
Again, this is an invalid proof. To show that is it invalid, let me use the same technique to “prove” that 2 equals 1.
Prove:
\[ 2 = 1 \]
The symmetric property of equality gives
\[ 1 = 2 \]
And, because I can add equal things to equal things,
\[ 2 + 1 = 1 + 2 \]
Subtracting 2 from each side,
\[ 1 = 1 \] which is obviously true. QED, we have shown that $2 = 1$.
The problem with these “proofs” is that they start by asserting the very equality that we want to prove. If that assertion is, in false, we can still prove all manner of true (and false) conclusions from it.
How should you structure such a proof? Try:
\[ \begin{align} \mbox{expression}_1 & = \mbox{expression}_{1a} \\ & = \mbox{expression}_{1b} \\ & = \mbox{expression}_{1c} \\ & \vdots \\ & = \mbox{expression}_{2} \\ \end{align} \]
where each right-hand side is derived from the expression before it by some obviously true manipulation.
For example:
Prove that, if $\sum_{i=0}^{k} 1 = k + 1$ then it must be true that $\left(\sum_{i=0}^{k+1} 1\right) = k + 2$.
Expanding the sum:
\[ \left(\sum_{i=0}^{k+1} 1\right) = \left(\sum_{i=0}^{k} 1\right) + 1 \]
We have stipulated that $\sum_{i=0}^{k} 1 = k + 1$, so
\[ \begin{align} \left(\sum_{i=0}^{k+1} 1\right) & = (k + 1) + 1 \\ & = k + 2 \\ \end{align} \]
QED
Notice that the left-hand side of the equalities we wrote never changed.
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