In this lesson, we will learn how to prove mathematical statements by deductive reasoning or by exhaustion, or disprove them using a counterexample.

Students will be able to

Q1:

Liam wants to prove the statement that π₯+2π₯+3ο¨ is positive for any real value of π₯. He does the following calculation:

π₯+2π₯+3>0ο¨

(π₯+1)β1+3>0ο¨

(π₯+1)+2>0ο¨.

Since (π₯+1)β₯0ο¨ for all real values of π₯, the statement is true.

Identify a potential problem with Liamβs calculation.

Q2:

Olivia wants to prove that all the cube numbers between 1 and 100 (inclusive) can be written in the form 9πβ1, 9π, or 9π+1, where π is an integer.

If she wants to prove this by exhaustion by checking each number directly, how many numbers will she need to check?

Olivia now wants to prove the statement for all possible cube numbers by considering πο© for different cases of π. Is this possible? If so, which cases of π would she need to check?

Q3:

Identify the counterexample to this statement: There is no positive integer other than 6 that is equal to half the sum of its positive divisors.

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