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:
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:
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 $\,2$ |
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.
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 |
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 |