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 higherlevel mathematics, proofs are usually written in paragraph form.
When introducing proofs, however,
a twocolumn 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 twocolumn 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 twocolumn 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 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
