Writing an Indirect Proof
Start with two possible statements. Indirect proofs work if you can describe the situation in two possible ways. Since there are only two options, once you prove one statement wrong, you will know the other one is correct. These are usually just two opposites: "A is true" and "A is not true." 'Example:' Think of a suspect in a police investigation. There are two possible explanations: the suspect is innocent; or the suspect is guilty. If we can rule out the idea that he's guilty, we automatically know he is innocent.
Write down what you know is true. These statements are often called "axioms" or "givens" (as in, the information given to you). You don't have to write down every fact you know, but it may help to write down related, proven statements. These can help you draw logical conclusions. Example: "The person who committed the crime was at the crime scene." and "A person cannot be in two places at once." are two examples of real-life "givens." These should be so obvious you can include them in your proof without needing evidence.
Assume one of the statements is true. Pick the one you think you can disprove most easily. Start with the idea "what if this statement is actually true?" This is called a postulate. The goal of the indirect proof is to show where this postulate leads. Example: Assume the suspect is guilty. It might not be true, but that's what this proof will tell us.
Draw logical conclusions and look for contradictions. Why is it useful to assume something that might not be true? The goal is not to find out the truth, but to look for contradictions. If your assumption leads to two contradictory statements, or if it contradicts one of your "givens," it means your assumption must be wrong. Example: If the suspect is guilty, as you've assumed, he must have been present while the crime was committed.Witnesses saw the suspect in a different city on the day of the crime.These two facts contradict each other. If you fail to find any contradictions, it does not mean your assumption was correct, only that it is possible.
Conclude that your assumption was incorrect. If you found a contradiction, and there are no faults with your logic, your initial assumption must have been wrong. Example: The suspect could not have committed the crime and been in a different city at the same time. Therefore, the assumption that the suspect was guilty must be incorrect.
Infer that the other statement must be correct. Now you know one statement is incorrect. Since there is only one other possible statement, that one must be right. You have now proved this statement indirectly. Example: Since the suspect cannot be guilty, he must be innocent. Notice that you do not need to spend any time investigating the other statement.
Mathematical Indirect Proof: a Triangle Cannot Have More Than One Right Angle
List the two possibilities. Here's a more mathematical example. The two statements are "A triangle can have more than one right angle" and "A triangle cannot have more than one right angle." Only one of these statements can be correct.
Set up the given information. In this case, the information needed for this proof is "the sum of all angles in a triangle is 180 degrees." This is usually proven earlier in the math textbook, or provided as a truthful statement.
Assume a triangle can have more than one right angle. This is the statement that seems easiest to disprove, so this is where to start. Imagine a triangle with two right angles (angles a and b), and one unknown angle (angle c).
Sum the two right angles. Each right angle is 90 degrees. Angles a and b are both right angles, so a + b = 90 + 90 = 180 degrees.
Try to find the value of the unknown angle. Our given information states that all three angles add up to 180 degrees. This means angles a + b + c = 180 degrees. Solve for c: 'a + b + c = 180 We already found that a + b = 180, therefore 180 + c = 180. c = 180 - 180 = 0.
Look for contradictions. The solution that angle c is 0 degrees is impossible, since a triangle with a zero degree angle is impossible.
Draw conclusions. Since you found a contradiction, the assumption "a triangle can have more than one right angle" must be false. Therefore, by indirect proof, the other statement must be correct. A triangle cannot have more than one right angle.
Comments
0 comment