Tuesday, September 15, 2015

Lesson 2-5: Good Definitions (Day 14)

This is what I wrote last year about today's lesson. I updated some information that I said about the corresponding section in Dr. Franklin Mason's text, because he has since made a change:

Lesson 2-5 of the U of Chicago text deals with definitions -- the backbone of mathematical logic. Many problems in geometry -- both proof and otherwise -- are simplified when students know the definition.

Consider the following non-mathematical example:

Given: My friend is Canadian.
Prove: My friend comes from Canada.

The proof, of course, is obvious. The friend comes from Canada because that's the definition of Canadian -- that's what Canadian means. But many English speakers don't think about this -- if I were to say to someone, "My friend is Canadian. Prove that my friend comes from Canada," the thought process would be, "Didn't you just tell me that?" Most people would think that "My friend is Canadian" and "My friend is from Canada" as being two identical statements -- rather than two nonidentical statements that are related in that the second follows from the first from the definition of Canadian. Yet this is precisely how a mathematician thinks -- and how a student must think if he or she wants to be successful in mathematics.

And so, let's take the first definition given in this section -- that of midpoint -- and consider:

Given: M is the midpoint of AB
Prove: AM = MB

The proof is once again trivial -- AM = MB comes directly from the definition of midpoint.

The text proceeds with the definition of a few other terms -- equidistantcircle, and a few terms closely related to circles. Then the text emphasizes biconditional statements -- that is, statements containing the phrase "if and only if." Some mathematicians abbreviate this phrase as "iff" -- but very few textbooks actually use this abbreviation.

Notice that Dr. Franklin Mason does give the "iff" abbreviation in his text. Last year, I noticed that his Lesson 2.4 on biconditionals had an (H) symbol -- which stands for honors. It's interesting how Dr. M once considered this to be an honors-only topic. Now, he wants to emphasize the importance of definitions and biconditionals to all his students, not just his honors students.

Every definition, according to the U of Chicago text, is a biconditional statement, with one direction being called the "meaning" and the other the "sufficient" condition. Mathematicians often use the terms "necessary" and "sufficient." Many texts use the word "if" in definitions when "if and only if" would be proper -- but our U of Chicago text is careful to use "if and only if" always with definitions.

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