INTRODUCTION TO THE TWO-COLUMN PROOF

• Jump right to the exercises!
• Before studying this section, you may want to review:
  Practice with the mathematical words ‘and’, ‘or’, and ‘is equivalent to’
  ‘If... Then...’ Sentences
  Contrapositive and Converse
  Proof Techniques
  Logical Equivalences and Practice with Truth Tables

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 and tautologies (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 be proving very simple algebraic statements—the goal is to practice with structure and style, and not be distracted by difficult content.
You will also practice with the methods of direct proof, indirect proof, and proof by contraposition.

Here are your first two-column proofs:

PROVE:

If [beautiful math coming... please be patient] $\,2x + 1 = 7\,$, then [beautiful math coming... please be patient] $\,x = 3\,$.
Use a direct proof.

PROOF:

STATEMENTS	REASONS
1.   Assume: $\,2x + 1 = 7\,$	hypothesis of direct proof
2.   $2x = 6$	Addition Property of Equality; subtract $\,1\,$ from both sides
3.   $x = 3$	Multiplication Property of Equality; divide both sides by $\,2$

PROVE:

If [beautiful math coming... please be patient] $\,2x + 1 = 7\,$, then [beautiful math coming... please be patient] $\,x = 3\,$.
Use an indirect proof.

In this case, an indirect proof is much longer than a direct proof.
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 $\,x\ne 3\,$	hypothesis of indirect proof
2.   $2x + 1 = 7$	$(A\text{ and }B)\Rightarrow A$
3.   $2x = 6$	Addition Property of Equality; subtract $\,1\,$ from both sides
4.   $x = 3$	Multiplication Property of Equality; divide both sides by $\,2$
5.   $x \ne 3$	$(A\text{ and }B)\Rightarrow B\,$   (step 1)
6.   $x = 3\,$ and $\,x\ne 3\,$; CONTRADICTION	(steps 4 and 5)
7.   Thus, $\,x = 3\,$.	conclusion of indirect proof

PROVE:

If [beautiful math coming... please be patient] $\,2x + 1 = 7\,$, then [beautiful math coming... please be patient] $\,x = 3\,$.
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: $\,x\ne 3\,$	hypothesis of proof by contraposition
2.   $2x \ne 6$	Multiplication Property of Equality; multiply both sides by $\,2$
3.   $2x + 1 \ne 7$	Addition Property of Equality; 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.