This is the second of our three WitCHes on VCAA’s Specialist Mathematics Exam 1 Sample Questions. It may not be quite bad enough for a WitCH, but it’s very not good. Our main reason for posting it is because we believe there is more in the question than might meet a teacher’s eye. As with our previous WitCH, this question is on a new topic, and some visible discussion seems worthwhile.
No one ever pays attention to the titles. Anyway, on to the question, which is a disaster (and is worded horribly).
As many commenters have noted, below and here, there is absolutely no point in a proof by contradiction, since the question can easily and naturally be proved directly. Plus, the proof by contradiction highlights an issue, which VCAA ignores but which is worth at least some consideration, and which highlights a(nother) major VCAA failing.
First the direct proof, and for this let’s make clear our definitions: an integer n is even if we can write n = 2k for another integer k; and, n is odd if we can write n = 2k + 1 for an integer k. This stuff may seem “well, duh”, but, as we will discuss below, it is not.
Now, for the direct proof, we are assuming n is odd, which means we can write n = 2k + 1. So,
n3 + 1 = (2k + 1)3 + 1 = 8k3 + 12k2 + 6k + 1 + 1 = 2(a bunch of integer stuff).
What about a proof by contradiction? Well, that requires assuming there is an odd integer n for which the conclusion “n3 + 1 is even” is false, which would mean n3 + 1 is odd. Is that possible? Well, remembering that n is odd, we can write n = 2k + 1. Then, calculating exactly as we did in the direct proof, we find that n3 + 1 = 2(integer stuff), which is therefore even. But we assumed n3 + 1 was odd, and so we have a contradiction. Done.
There are two points to make about this proof by contradiction. The obvious point is that there is no point. The fundamental content of the proof is identical to that of the direct proof. The contradiction proof simply cloaks the direct proof in a layer of gratuitous and confusing logic. (Incredibly, the webinar makes the contradiction proof even worse.)
The second point to make is that the contradiction proof above is fallacious. Remember that “well, duh” stuff above?
In the proof by contradiction, we assumed the conclusion “n3 + 1 is even” is false. But the negation of that statement is “n3 + 1 is not even“. Now, does an integer being “not even” imply that it is odd? Yes, but it takes a proof.
What is missing from the proof by contradiction above is a proof of, or at least a clear statement that, every integer is either even or odd. This can done easily, and nicely, by induction.
Should students simply be able to assume that “not even” means odd? How would they know? Should they then then also be able to assume ODD x ODD = ODD? If so, there is an even simpler direct proof of the n3 + 1 thing.
When setting out proof as a topic, it is a fundamental responsibility to clearly lay out the ground rules. VCAA has, instead, provided a swamp.