Introduction to the Two-Column Proof
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 is 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 $\,2x + 1 = 7\,,$ then $\,x = 3\,.$ 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 $\,2x + 1 = 7\,,$ then $\,x = 3\,.$ 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 $\,2x + 1 = 7\,,$ 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: $\,x\ne 3\,$ | 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 |
Concept Practice
This is an optional introductory paragraph.