Sunday, August 24, 2014

Section 2-5: Good Definitions (Day 13)

Section 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 -- equidistant, circle, 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. I also notice that his Section 2-4 on biconditionals has an (H) symbol -- which stands for honors. It seems interesting that Dr. M would consider this an honors-only topic -- but then again, we, as teachers, don't necessarily want the students to be bogged down in formalism and proofs.

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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