Lesson: Mathematical Logic and Proof Mathematics
In this lesson, we will learn how to prove mathematical statements by deductive reasoning or by exhaustion, or disprove them using a counterexample.
Liam wants to prove the statement that is positive for any real value of . He does the following calculation:
Since for all real values of , the statement is true.
Identify a potential problem with Liam’s calculation.
Olivia wants to prove that all the cube numbers between 1 and 100 (inclusive) can be written in the form , , or , where 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 for different cases of . Is this possible? If so, which cases of would she need to check?
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.