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Honors Geometry Textbook24 chapters | 200 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson defines both direct and indirect proofs and, in turn, points out the differences between them. We'll also look at some examples of both types of proofs in both abstract and real-world contexts.

Suppose you and your friend Rachel are going to an art festival. When you get there, you are the only ones there. Rachel looks at you and says, ''If the art festival was today, there would be hundreds of people here, so it can't be today.''

You take out your tickets, look at the date and say, ''The date on the tickets is for tomorrow, so the art festival is not today.''

Notice that both you and Rachel came to the same conclusion, but you got to that conclusion in different ways. As it turns out, your argument is an example of a direct proof, and Rachel's argument is an example of an indirect proof.

- A
**direct proof**assumes that the hypothesis of a conjecture is true, and then uses a series of logical deductions to prove that the conclusion of the conjecture is true.

- An
**indirect proof**relies on a contradiction to prove a given conjecture by assuming the conjecture is not true, and then running into a contradiction proving that the conjecture must be true.

The definitions of direct and indirect proofs give way to the steps we follow to perform each type of proof.

To perform a direct proof, we use the following steps:

- Identify the hypothesis and conclusion of the conjecture you're trying to prove

- Assume the hypothesis to be true

- Use definitions, properties, theorems, etc. to make a series of deductions that eventually prove the conclusion of the conjecture to be true

- State that by direct proof, the conclusion of the statement must be true

Consider your arguments again. In your argument (direct proof), you use the fact that the tickets say that the art festival is tomorrow to prove that the art festival can't be today. You use a direct proof by using logical deductions to prove a conclusion.

But to perform an indirect proof, we use a different process which includes the following steps:

- Assume the opposite of the conjecture, or assume that the conjecture is false

- Try to prove your assumption directly until you run into a contradiction

- Since we get a contradiction, it must be the case that the assumption that the opposite of the hypothesis is true is false

- State that by contradiction, the original conjecture must be true

In Rachel's argument (indirect proof), she starts by assuming the opposite of the original conjecture, which is that the festival is not today. That is, she starts with ''If the art festival was today'', then she says, ''there would be hundreds of people here.''

This is a contradiction, since you and Rachel are the only ones there. Lastly, she concludes that ''the art festival can't be today.'' All together, she uses in indirect proof by assuming the opposite of the conjecture, identifying a contradiction, and stating that the original conjecture must be true.

Okay, now that we understand direct and indirect proofs, let's get a bit more mathematical. Suppose we want to prove the following statement:

- The number 7 is a rational number.

First, let's consider proving it directly.

**Direct Proof:**

A rational number is defined as a number that can be written in the form *p*/*q*, where *p* and *q* are integers. The number 7 can be rewritten as 7/1, because 7 divided by 1 is still 7. Since 7 and 1 are both integers, and 7 can be written as 7/1, we have that by the definition of a rational number, 7 is a rational number.

Now, let's see what happens if we prove it indirectly.

**Indirect Proof:**

Assume that the number 7 is not a rational number. A number that is not rational is called irrational and cannot be written as a fraction, *p*/*q*, where *p* and *q* are both integers. Since 7 is not rational, it must be irrational. However, we can write 7 as 7/1, where 7 and 1 are integers, but it's impossible to write irrational numbers as fractions. This is our contradiction, so 7 must be a rational number.

In this instance, the direct proof is a little shorter and easier to use. However, there are many instances when an indirect proof is easier.

For example, suppose we wanted to prove the statement:

- If
*a*+*b*is odd, then*a*or*b*must be odd.

If we try proving this directly, we will find it to be quite tricky. However, take a look at if we prove it indirectly:

**Indirect Proof:**

Assume that *a* + *b* is odd, but that neither *a* nor *b* are odd. Then *a* and *b* must both be even. By the definition of an even number, *a* = 2*k* and *b* = 2*m*, where *k* and *m* are integers. Therefore,

*a*+*b*= 2*k*+ 2*m*= 2(*k*+*m*)

Since the integers are closed under addition, *k* + *m* is an integer, so *a* + *b* is even by the definition of even numbers, but by assumption, *a* + *b* is odd. This is our contradiction. By contradiction, it must be the case that if *a* + *b* is odd, then *a* is odd or *b* is odd.

Easy, right? The indirect proof is the more appropriate method in this case.

Let's reviewâ€¦

- A
**direct proof**assumes that the hypothesis of a conjecture is true, and then uses a series of logical deductions to prove that the conclusion of the conjecture is true.

- An
**indirect proof**relies on a contradiction to prove a given conjecture by assuming the conjecture is not true, and then running into a contradiction proving that the conjecture must be true.

Direct and indirect proofs are used quite often in mathematics, and each of them lends itself to proving statements in unique ways. The more we practice these types of proofs, the more we will be able to identify which proof method is most appropriate for a given situation. So, let's keep in mind the old saying that practice makes perfect and try to work with both types of proofs as much as possible!

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Honors Geometry Textbook24 chapters | 200 lessons

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