Math 323 Comments on Proofs

On proving things (three examples are given below):

Suppose a theorem to be proved states:

If [assumptions], then [conclusion].
Students often try to prove such a theorem in one of two ways.
  1. They state [conclusion], do some manipulations on [conclusion], and end up with [assumptions].

    OR

  2. They state [assumptions], do some manipulations on [assumptions], and end up with [conclusion].
When using method (1), students may perhaps go so far as to start with Then, they do some manipulations on [conclusion] and end up with either [assumptions] or some trivial statement such as “1 = 1” -- at which point they gleefully assert, “which is true”, and mistakenly think that they have proved what is supposed to be proved.

As pointed out several times in class, and as pointed out when papers are graded, and as also pointed out in the textbook, this is WRONG!

There is nothing logically or mathematically wrong with the second approach (which I call the “plop it down” method, because you begin by just “plopping down” a true statement, hoping that you can get to where you want to go). It can be used on very simple problems. As soon as one gets past very simple problems, however, this is usually not the best strategy to use.

The best approach (usually) is not to state the assumptions and (try to) manipulate them into the conclusion, and certainly not to state the conclusion and (try to) manipulate it into the assumptions. Instead, one wants to examine the conclusion and try to use the assumptions to get from “one side” of the conclusion to the “other side” (which I call the “follow it through” method).

Below is a good, simple illustration of the three approaches to a situation which students who have completed a year of calculus should be familiar with. This is not an example you can slavishly follow in your proofs; it is merely an illustration of the point.

Examples of Proof Methods

(Click here or here to open a graphical image of these examples in a new window.
Click here for a pdf file with these examples in a new window.)

THEOREM.
      Assumptions:

c2 + s2 = 1;   c is not 0;   by definition, t = s/c.

      Conclusion:

1 + t2 = (1/c)2.

(Think of cos, sin, and tan.)

Proof, Method I


1 + t2 = (1/c)2
1 + t2 = 1/ c2   (algebra)
c2 + (c2)(t2) = 1   (multiply by c2).
c2 + (c2)(s/ c)2 = 1   (definition of t )
c2 + s2 = 1   (algebra)
Using one of the assumptions, we get
      1 = 1         WHICH IS TRUE!
THIS IS BAD! UPSIDE DOWN!

Proof, Method II (“plop it down” method)

c2 + s2  =  1   (assumptions)
         (Start with one of your assumptions -- hope it is the right one.)
1 +   s2/ c2  =  1/ c2   (divide both sides by c2 )
       (Do something to the assumption you started with -- hope it is the right thing to do.)
1 +  (s/ c)2  =  (1/c)2   (algebra)
1 +  t2  =  (1/ c)2   (definition of t ).

This is correct, but it takes a little thought to figure out which assumption to start with and the first step after stating the assumptions, and the following approach is usually better:

Proof, Method III (“follow it through” method}

We want to prove 1 + t2 = (1/c)2.
Then (start with something specific -- one side of the statement you are trying to prove.)
1 + t2 = 1 + (s/c)2   (definition of t)
      = 1 + s2/c2   (algebra)
      = (c2 + s2)/c2   (algebra; common denominator)
      = 1/c2   (by assumptions)
      = (1/c)2

(Usually, in the proofs of the type done in our course, one uses more of a sentence and paragraph structure than the tabular "assertion-reason" format illustrated here.)


Analogy: In certain games--e.g., basketball, football, hockey, soccer--one very often has the ball (or puck) at one end of the playing field and one wants to get it to the other end. (This is analogous to getting from 1 + t2 to (1/c)2 in the preceding example.) How does one do this? Does one take a time out, read the rule book (analogous to the definitions in math), read the scout's report on the other team and note the specific location of the ball (analogous to the assumptions in mathematics), and try to deduce from these a complete plan to go from one end of the field to the other? Not usually. Instead, one starts at one end (1 + t2) and uses the rules and circumstances (definitions and assumptions) ALONG THE WAY to try to get to the other end ((1/c)2).


Return to Course Home Page.