Introduction

Last year, you discovered that whenever you inscribed a triangle in a circle with one side being a diameter of the circle, the triangle seemed to be a right triangle. And some of you might have concluded, not without good reason, that this must always be the case. This kind of reasoning is called $inductive reasoning$— when you use a set of examples to claim a general theory.

Most of you, however, may have remained unconvinced that this would be always the case, and set out to prove it using some of the axioms of geometry. The kind of reasoning you used is called $deductive reasoning$— the kind of reasoning preferred by mathematicians, and the kind which will be the focus of this lesson.

In mathematics, inductive reasoning is often used to make a guess at a property by examining some cases, and deductive reasoning (deduction) to prove that the property generally holds.

As in the case of the circle example above the conclusion reached by inductive reasoning proved to be indeed correct, but in many cases, conclusions reached by inductive reasoning will turn out to be false. If you check the value of the expression for the first 10 or so positive integer values of $n$, you might conclude that was a prime number for any positive integer $n$. It turns out that this is not the case.

Can you find a value for n such that the expression is not a prime number?

In this lesson not only will we explore logic problems including some similar to the ones in problem 1, but we will also introduce a different kind of geometry, called $finite geometry$. As its name suggests, finite geometry, unlike the geometry to which you are accustomed with an infinity of points, lines, planes, and so on, concerns itself solely with finite numbers of elements.

We will study some of these geometries not only because of their quirky elegance but also because they provide a look into the nature of mathematical proof and help to sharpen your skills in developing valid logical arguments. They also show the power of visualization as an aid in solving more difficult problems.

Development

As you know from your work in Euclidean Geometry, any system of deductive reasoning consists of undefined terms, definitions, postulates (axioms), rules of logic (akin to common sense), and theorems (proved statements).

Recall that a $postulate$($axiom)$ is a statement whose truth is to be accepted without proof, such as the line postulate in Euclidean Geometry, which states that one and only one line can be drawn through 2 distinct (different) points.

And the truth of a $theorem$ is established by applying the rules of logic to the postulates. At each step in the proof of a theorem you must indicate which postulate you are applying, or which previously proved theorem you are drawing upon.

Why must there be undefined terms? Is it possible to define every word/term? Try defining the color blue for a person who has no idea what blue is. And assume that if a person does not know what blue is, it is likely that the person will not know a host of other words associated with the color blue.

But, of more importance, would the correctness of your reasoning be in question if some of the words used in the postulates were not defined? For example, supposed we replaced some of the words in problem 1a with words whose meanings you did not know, would you still conclude that it was a case of deductive reasoning?

There are a few different ways to solve problem 4. You might have used a kind of indirect argument, or, if you are more visually inclined, you might have used a Venn diagram, as suggested by problem 3.

This kind of argument you used in problem 5 is called $proof by contradiction.$ Here is how it works. First, assume that the original statement you want to prove is false. Second, by reasoning from this assumption, reach a conclusion that is impossible — impossible, either because it contradicts a postulate or an established theorem, or because it contradicts the given information. Third, since assuming your original statement was false led to contradictions of known facts, you can now conclude that the original statement must instead be true.

This kind of argument, proof by contradiction, makes use of a very important fact in the rules of logic — that any statement is equivalent to (exactly the same as) its $contrapositive$. Some of you may have encountered this term before. Here’s an example of a statement and its contrapositive.

Statement: To be selected for Eightnotes you have to be a singing sensation.

Contrapositive: If you are not a singing sensation, then you will not be selected to Eightnotes. [or you can kiss Eightnotes goodbye.]

Here’s another:

Statement: If an animal is a zebra, then it has stripes.

Contrapositive: If an animal does not have stripes, then it is not a zebra.

Would you agree that in the example above the statement and its contrapositive are equivalent? What follows is a somewhat less subtle use of the contrapositive. In the problem, x is considered to be an integer.

Back to problem 4. For a more visual approach, you might have represented each postulate by a Venn diagram (see below) and used the final picture to reach your conclusion.

$Postulate 1 $

$Postulate 2$ $Postulate 3$

Next we will take a look at some finite geometries. But before we explore a more traditional example with “points”, “lines”, etc., let’s look at one that might be a tad more familiar.

Here is a more traditional one with three postulates. The terms “lines” and “points” are undefined and you must resist the temptation to think of these elements the way you did in Euclidean Geometry. For example, in a finite geometry a “line” cannot contain an infinite number of “points” since there are only a finite number of “points” in these geometries.

The following is also a model for the system, but is essentially the same as the one above. Why does this claim make sense? (Think back to your graph theory lesson.)

Although models are extremely helpful in creating and examining conjectures, you have to be careful not to rely on models to construct proofs. In a very few cases it might be possible to construct every possible model satisfying all the postulates, but in general that would be very difficult to do. And even if you could construct all possible models, it might still be difficult to prove that you have done so. So to prove conclusions, you should reason directly from the postulates.

Before working in any deductive system it is important to examine the postulates for $consistency$— is it possible for all the postulates to be true at the same time? The rule of thumb is that if you can draw a diagram to represent the set of postulates, then the postulates are consistent. Can you come up with some reasonable argument why drawing a diagram would allow the claim that the postulates are consistent? The systems in problems 8 and 9 passed the test since we were able to construct a model for each.

Practice