Below are three statements together with a possible first...

Below are three statements together with a possible first line of a proof of that statement. In each case, say whether the first line is the start of a direct proof, a proof by contrapositive, or a proof by contradiction. 1. Statement: For any triangle, the sum of the interior angles is 180 degrees. First line: Consider a plane figure and assume that the sum of the interior angles is not 180 degrees. 2. Statement: For every integer n, the number 7n - 1 is divisible by 6. First line: Suppose there were some integer n for which 7n - 1 was not divisible by 6. 3. Statement: Every three digit number has a sum of digits at most 27. First line: and assume n has three digits.

Transcript

00:01 Okay, in this problem, we have these three statements, and together with a possible first line of a proof of that statement.

00:06 And we have to say what kind it is.

00:08 Is it a direct proof, a proof by contrapositive, or proof by contradiction? okay, so what we're going to do is a direct proof is going to be just we're going to start with our premise and then just go directly to our conclusion.

00:20 So that's always going to be just let my premise be true.

00:24 All right.

00:25 That's the most basic form of proof.

00:28 Now, the next one is the contraposal.

00:30 And that means i'm going to basically negate and flip this statement so that instead of saying let my premise be true, i'm going to say suppose that my conclusion is false.

00:40 So let my conclusion be false and then we're going to show that my premise must be false as well.

00:46 And that's going to be the same.

00:47 Now contradiction is going to be kind of i like to say as suppose not.

00:53 Suppose that i'm wrong.

00:54 And that's going to be like, well, suppose that my premise is true.

00:57 And then we're going to assume that my conclusion.

01:00 Is false.

01:01 So we're going to say, yeah, my premise is through and my conclusion is false.

01:07 So we're going to assume that my conclusion is false.

01:09 So with this first one for any triangle, the sum of the interior angles is 180.

01:14 All right.

01:15 So we say, if a triangle, then the angles add up to 180.

01:18 So now we're going to say, consider a plain figure and assume that the sum of the interior angles is not 180 degrees.

01:27 So notice that keyword.

01:29 We have that keyword.

01:30 We have that key word.

01:30 We have that key not, all right? we're going to have that keyword be not...

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