Indirect Proofs

Integer Proofs

Indirect Reasoning may be used in proofs.Sometimes , there are situations where it is not possible to directly prove a statement.

But we can sometimes prove that it cannot be false.

In an indirect proof we assume that the statement we wish to prove true is actually false.With this assumption we continue our reasoning until we reach a conclusion that condradicts something we know to be true.This tells us that our assumption was false, and therefore the opposite assumption is true.

In cases where there are only two possiblities, the proof is complete.

Steps For Writing an Indirect Proof

1) State all possiblities

2) Assume the opposite of what you want to prove is true.

3)Reason correctly from the given information until a contradiction of a known postulate, theorem, or given fact is reached.(Deductive Reasoning)

4) State that what was assumed to be true in Step 2 is false.Hence, it follows that one of the other possibilities is true.

5) Eventually the one remaining possibility is the desired conculsion.In the case where there are only two possiblities our proof is complete.

Example 1

We often use indirect reasoning in our everyday lives.Take for example, that you and a friend are going to a Matchbox 20 Concert. When you arrive you find that you and your friend and two co-workers are the only ones there.You are pretty convinced by now that the concert is not going to be tonight.Your reasoning might sound something like this:"If this were the concert day, there would be hundreds more people here, and we are the only ones here.Therefore, this cannot be the concert day."

Example 2

Consider this situation:

Marie,a lawyer, was defending her client, CP Rail, in a lawsuit launched by Jeff Miller over alleged injuries incurred in a train accident outside Toronto.Miller claimed that while he was sleeping in his berth, the train had slammed on its brakes while backing up causing his head to strike the wall behind his pillow.He claimed he had sustained a skull fracture causing him great stress on the job and in his personal life, due to headaches.Miller was suing CP Rail for two million dollars.Marie presented the defense's case saying," Assume Mr. Miller is telling the truth. On the night when the accident occurred, the porter made up all the berths in the railway car with the heads toward the front of the train. The train was backing up when it came to its sudden stop. The force of the braking would cause Mr. Miller's feet to strike the wall, not his head.Therefore, Mr.Miller is not telling the truth."CP Rail won the case.

Marie followed the steps of indirect reasoning.

1) She knew that either Mr.Miller was lying or he was telling the truth.

2) She assumed the opposite of what he was attempting to prove was true; that Mr. Miller was telling the truth.

3) He followed this assumption with a series of deductive statements until he reached a contradiction.

4) Therefore the assumption was false.

5) The other possibilty must be true - that Miller was lying.

Therfore, in an indirect proof, we are assuming that the desired conculsion is false.If this assumption leads to a contradiction of a known fact, then we can conclude that our assumption was false and that the desired conclusion is true.

Example 3

Now we will try an example using indirect proofs in geometry.

Example 3 : Using Indirect Proofs in Geometry

Plan: Draw the diagram.Use an indirect proof. Either the triangles are congruent or they are not.Assume that the triangles are congruent and reason to a contradiction.

Triangle Graphic