Comments on Test 4

below the fold... this is to help with the Final Exam preparation.

• Quite a few people do not seem to understand what countable means, when we talk about a countable set. (The video from referred to "listable set", meaning the same thing, but that name is not standard.)
Please see this post from the class where it was discussed.

• When stating the Pythagorean Theorem (or any theorem, for that matter), it is important what you say and what you don't say. I am not a big advocate of just memorizing sentences you don't understand, because that never works out well: usually it's difficult to impossible to remember something accurately if you don't really understand what it means. Think of all the ways people (especially young children) mangle things like the Pledge of Allegiance.

If you understand what the statement of the theorem is saying, you should be able to say it in slightly different words without making it into something else.

For the Pythagorean Theorem, there are three important parts (and I did say this when I discussed it in class!): It only refers to right triangles, that is, triangles that have a right angle, so you have to mention that; the hypotenuse (or you could call it that side opposite the right angle) plays a special role, so you have to mention its name; and the theorem says that two things are equal, namely, the square of the hypotenuse, and the sum of the squares of the other two sides.

As long as you've covered those, understand what they mean, and you've correctly stated the conclusion, you are in good shape.

Just trying to memorize the words can go terribly wrong. "The sum of the squares" is not the same thing as "the square of the sum" for example.

The same thing applies to the proof of the theorem, or any proof of any theorem. Just trying to memorize sentences or lists of equations (I have in mind the proof that the square root of 3 is irrational) can go horribly wrong if you do not understand what the sentences or equations are saying, and how they are logically related to each other. That is one reason that I went back and outlined the lines of reasoning in some of our big proofs.

A really important thing you should understand from reading and studying proofs in this courseis how they prove what they claim to prove: How you can prove a general statement is always true. Many students seemed to think that they could "prove" the Pythagorean Theorem by using a specific example, namely (usually) the 3-4-5 right triangle. This fails for two reasons: first, because in mathematics we can never prove a general statement by looking at one example, or even at several examples. (And "prove" does not mean "test", by the way. It means "show that this statement is true in all generality.")

The second reason that using the 3-4-5 triangle fails as a "proof" of the Pythagorean Theorem is even more important: We only know that the 3-4-5 triangle has a right angle because of the Pythagorean Theorem (or, rather, its converse). Otherwise there is no way to know that the angle in this triangle is truly a right angle and not just very-very close to a right angle!

SO now go back and look at the "algebraic" proof and see how it makes no assumption at all except that the triangle has a right angle. The proof does use the fact (it's another theorem) that all the angles of a triangle add up to a straight line; that's not a problem, because that theorem has already been proved.