What’s the difference between a direct proof and an indirect proof?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
That’s exactly what you’re going to learn in today’s discrete lecture.
Let’s dive right in!
Direct Proof Definition
Well, as we learned in our previous lesson, a direct proof always assumes the hypothesis is true and then logically deduces the conclusion (i.e., “if p is true, then q is true).
Indirect Proof Definition
An indirect proof doesn’t require us to prove the conclusion to be true. Instead, it suffices to show that all the alternatives are false.
There are two forms of an indirect proof.
- Proof By Contraposition
- Proof By Contradiction
Proof By Contrapositive
Okay, so a proof by contraposition, which is sometimes called a proof by contrapositive, flips the script. Instead of assuming the hypothesis to be true and the proving that the conclusion is also true, we instead, assumes that the conclusion to be false and prove that the hypothesis is also false.
Remember, we know from our study of equivalence that the conditional statement of “if p then q” has the same truth value of “if not q then not p.” Therefore, a proof by contraposition says, let’s assume “not q” is true and let’s prove “not p.” And consequently, if we can show “not q then not p” to be true, then the statement “if p then q” must be true also as noted by the State University of New York.
Example #1
For instance.
Contraposition Inequality Proof
Now I want to draw your attention to the critical word “or” in the claim above. Here’s a BIG hint…
…whenever you are given an “or” statement, you will always use proof by contraposition.
Why?
Because trying to prove an “or” statement is extremely tricky, therefore, when we use contraposition, we negate the “or” statement and apply De Morgan’s law, which turns the “or” into an “and” which made our proof-job easier!
Example #2
Let’s look at another problem.
Contrapositive Proof — Even and Odd Integers
Notice that by using contraposition, we could use one of our basic definitions, namely the definition of even integers, to help us prove our claim, which, once again, made our job so much easier.
Proof By Contradiction
Now it is time to look at the other indirect proof — proof by contradiction.
Like contraposition, we will assume the statement, “if p then q” to be false. In other words, the negation of p leads to a contradiction because if the negation of p is false, then it must true.
The steps for proof by contradiction are as follows:
- Assume the hypothesis is true and the conclusion to be false.
- Then show that this assumption is a contradiction, thus proving the original statement to be true.
Example #1
It may sound confusing, but it’s quite straightforward. Let’s look at some examples.
Contradiction Proof — N and N^2 Are Even
Not so bad, right?
Example #2
Let’s look at another one.
Proof By Contradiction — Not Equal
As you can see, it’s much easier to assume that something does equal a specific value than trying to show that it doesn’t. This is the beauty of the proof of contradiction.
Together, we will work through countless examples of proofs by contrapositive and contradiction, including showing that the square root of 2 is irrational!
Let’s jump right in.
Video Tutorial w/ Full Lesson & Detailed Examples
1 hr 43 min
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