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  • Use indirect reasoning to prove theorems about lines and angles
  • Use proof by contradiction and proof by contrapositive to prove conditional statements

Assignment

  • One vocabulary definitions
  • p62 15–22, 26–27 (PDF link)

The last section in the chapter has looks at proving claims in roundabout ways, rather than directly going from the given to the conclusion.

Proof by Contradiction

A contradiction is when you have two statements that both can’t be true. Saying that today is Thursday while also saying today is Friday is a contradiction. Both can’t possibly be true.

A proof by contradiction starts with assuming what you want to prove is false. For example, “Alani walked 8 kilometers the past two days, so she must have walked at least 4 kilometers on one of those days.”

Since the statement is claiming that Alani walked at least 4 kilometers on one of those days, let’s assume she didn’t do that. We’ll instead claim she walked less than 4 on both days.

Now we have a contradiction. If she walked 8 kilometers over two days, there’s no way to add two numbers and get 8 without one of them being at least 4. So, the original claim must be true.

For another example, we can prove that there is no “biggest number” by finding a contradiction. If we assume the opposite, that there is some number out there that bigger than all the others, we could easily add 1 to it to get a bigger number. So, we can’t have both a “biggest number” and the ability to add one to it, so there must not be a biggest number.

Proof by Contrapositive

Another way to indirectly prove something is to use its contrapositive. For any conditional to be true, its contrapositive must also be true. So, $\p\to q$ and $\neg q \to \neg p$ are both valid statements. For example, “if it’s raining, then I will bring my umbrella” and “if I didn’t bring my umbrella, then it is not raining”.

Depending on the conditional, the contrapositive might be easier to prove than the original.