Proof by Cases and Even/Odd Functions

The video tutorial introduces the proof by cases technique, a method used to prove a statement by considering multiple scenarios. The instructor explains the concept using an example: proving that n^3 - n is even for any integer n. The proof is divided into two cases: when n is even and when n is odd. For each case, the instructor demonstrates how to show that n^3 - n is even, concluding that the statement holds true in all scenarios. The tutorial emphasizes the importance of ensuring that all possible cases are covered in a proof by cases.

What is the main idea behind proof by cases?

• To prove a statement by direct computation.

• To prove a statement by considering all possible scenarios.

• To prove a statement by contradiction.

• To prove a statement by induction.

In proof by cases, what must be ensured about the cases considered?

• All cases must be true.

• At least one case must be true.

• None of the cases should overlap.

• All cases must be false.

What is the statement being proved in the example provided?

• n cubed plus n is odd.

• n cubed minus n is even.

• n cubed minus n is odd.

• n cubed plus n is even.

When n is even, how is n expressed in terms of k?

• n = k + 2

• n = 2k + 1

• n = 2k

• n = k^2

What is the common factor found in the expression for n cubed minus n when n is even?

• 5

• 2

• 4

• 3

When n is odd, how is n expressed in terms of k?

• n = k + 2

• n = k^2

• n = 2k

• n = 2k + 1

What is the result of n cubed minus n when n is odd?

• It is odd.

• It is even.

• It is a prime number.

• It is zero.

What is the significance of finding a common factor in the expression for n cubed minus n?

• It suggests that the expression is undefined.

• It proves that the expression is a prime number.

• It shows that the expression is divisible by that factor.

• It indicates that the expression is zero.

What conclusion is reached after considering both cases for n?

• n cubed minus n is always even.

• n cubed minus n is always odd.

• n cubed minus n is never even.

• n cubed minus n is sometimes even.

What is the purpose of using proof by cases in this example?

• To demonstrate that n cubed minus n is always even.

• To illustrate that n cubed minus n is zero.

• To prove that n cubed minus n is always odd.

• To show that n cubed minus n is a prime number.