Proof Techniques

Carol Fisher's Homepage
Geometry Table of Contents

For this exercise, you need ♥ INTERNET EXPLORER 6.0 and above, with MathPlayer installed.♥

PROOF TECHNIQUES

Jump right to the exercises!

If you need to review the mathematical words and, or, and is equivalent to, click here .

An identity is a mathematical sentence that is always true.
For example,   x+x=2x   is an identity.

A contradiction is a mathematical sentence that is always false.
For example,   x=1    and    x&neq;1   is a contradiction.

Given any mathematical statement   S ,
S  or  (not S)   is an identity, and
S  and  (not S)   is a contradiction,
as the truth table below shows:

S	not S	S  or  (not S)	S  and  (not S)
T	F	T	F
F	T	T	F

The proof by contradiction technique can be applied to any type of mathematical sentence:

PROOF BY CONTRADICTION:
To prove that a sentence is true, you can assume that it is false, and then reach a contradiction.

This approach is justified by the truth table below, where it is shown that S is equivalent to (not S)  ⇒  (a contradiction) :

S	not S	a contradiction	(not S)  ⇒  (a contradiction)
T	F	F	T
F	T	F	F

There are three standard approaches for proving an implication, A⇒B :

DIRECT PROOF:
To prove that an implication A⇒B is true,
we can assume that A is true, and show that B is also true.

The next method is based on the fact that an implication is equivalent to its contrapositive.

PROOF BY CONTRAPOSITION:
To prove that an implication A⇒B is true,
we can alternately prove that its contrapositive (not B)⇒ (not A) is true:
Assume that B is false, and show that A is also false.

Applying "proof by contradiction" to an implication leads to a proof technique that is called an "indirect proof":

INDIRECT PROOF:
To prove that an implication A⇒B is true,
we can assume that A is true, but B is false, and then reach a contradiction.

On this exercise, you will not key in your answers.
However, you can check to see if your answer is correct.

Please read my
TERMS OF USE