Introduction to the Two-Column Proof

Due to math content, this page has special requirements (including JavaScript) for full functionality.
With your current viewing scenario, it is not appearing and behaving as it is supposed to!
Please visit Dr. Carol J.V. Fisher's Homepage to learn what this site has to offer.
Watch the "Welcome" video to get started—hope to see you back here soon!

Dr. Carol J.V. Fisher's Homepage

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

Introduction to the Two-Column Proof

Jump right to the exercises!

Deductive reasoning uses logic, and statements that are already accepted to be true, to reach conclusions.
The methods of mathematical proof are based on deductive reasoning.

A proof is a convincing demonstration that a mathematical statement is necessarily true.
Proofs can use:

given information (information that is assumed to be true)
definitions (Definitions are true, by definition!)
postulates (statements that are assumed to be true, without proof)
logical equivalences (a truth table shows that these are always true)
statements that have already been proved

In higher-level mathematics, proofs are usually written in paragraph form.
When introducing proofs, however, a two-column format is usually used to summarize the information.
True statements are written in the first column.
A reason that justifies why each statement is true in written in the second column.

This section gives you practice with two-column proofs.
You will also get additional practice with the methods of direct proof, indirect proof, and proof by contraposition.

Here are your first two-column proofs:

PROVE: If 2x+1= 7 , then x=3 .
Use a direct proof.

PROOF:

STATEMENTS	REASONS
1. Assume: 2x+1= 7	hypothesis of direct proof (hyp of dir pf)
2. 2x= 6	Addition Property of Equality; subtract 1 from both sides (Add Prop of Eq; subtract 1 from both sides)
3. x= 3	Multiplication Property of Equality; divide both sides by 2 (Mult Prop of Eq; divide both sides by 2)

PROVE: If 2x+1= 7 , then x=3 .
Use an indirect proof.

In this case, an indirect proof is much longer than a direct proof.
Notice the abbreviations used for reasons.
Whenever you give a reason that uses anything except the immediately preceding step, then cite the step(s) that are being used.

PROOF:

STATEMENTS	REASONS
1. Assume: 2x+1= 7 AND x≠3	hypothesis of indirect proof (hyp of indir pf)
2. 2x+1= 7	(A and B)⇒A
3. 2x= 6	Add Prop of Eq; subtract 1 from both sides
4. x= 3	Mult Prop of Eq; divide both sides by 2
5. x≠3	(A and B)⇒B (step 1)
6. x= 3 and x≠ 3 CONTRADICTION	(steps 4 and 5)
7. Thus, x=3 .	conclusion of indirect proof

PROVE: If 2x+1= 7 , then x=3 .
Use a proof by contraposition.

In this case, the proof seems somewhat convoluted.
For this statement, a direct proof is best.

PROOF:

STATEMENTS	REASONS
1. Assume: x≠3	hypothesis of proof by contraposition (hyp of pf by contrapos)
2. 2x≠6	Mult Prop of Eq; multiply both sides by 2
3. 2x+1≠ 7	Add Prop of Eq; add 1 to both sides

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