Proof Methods Summary

By pukez4

Proof by Examples

Using a few examples is sufficient to prove a universally quantified statement

Example

Prove or disprove the statement:

  1. Taking , we find that
  2. Taking , we find that
  3. Taking , we find that

So the statement must be true.

Proof by Inspection

It looks like it's probably obviously true

Tip

Try being more assertive and confident with your proof. Since the marker will have (hopefully) marked at least 30 other exams beforehand, they are more likely to believe the proof to be trivial enough to be proven by inspection, especially if you seem confident. Fox e

E.g, "It is quite obvious after some use of the brain that this statement is true, and so by inspection"

Proof by Approximation

Also known as proof by engineering.

We can approximate certain variables so we can prove them with direct proof or inspection.

Theorem - The Fundamental Theorem of Engineering

Example

Prove or disprove the statement:

Proof by approximation: From the fundamental theorem of engineering, e know approximately, for small values, so

Once again, since we approximated above, we can use proof by approximation again to find that the statement is true, since 1 is close to 0.99

In fact, we can confirm this is true using proof by examples discussed above:

  1. Taking ,
  2. Taking ,
  3. The rest of the examples are trivial to find, and will be left as an exercise for the reader.

Proof by Multiplication of Zero

Multiplying both sides by 0 gives us a definitive way to prove or disprove a statement.

Example

Prove or disprove the statement:

We know: . Simplifying, we get , which satisfies the original inequality .

Proof by Necessity

It's on the exam (which I need to pass), and it says to prove, which implies it is true so it therefore must be true.

Note that this proof method does not work on questions that are ambiguous and say "prove or disprove".

Proof by Higher Authority

The TA said this question could indeed be proven, and since the proof is probably on an answer sheet somewhere, it implies that the statement is in fact true.

Once again, this does not work on questions that are ambiguous and say "prove or disprove". Why does this work? Well we don't prove the Pythagorean theorem every time we use it, because Mr. Pythagoras said it was true.