Proof by Exhaustion and Disproof by Counterexamples

The video tutorial covers proof by exhaustion and disproof by counterexample. It explains proof by exhaustion as a method to verify statements by checking all possible cases, using examples like finding prime numbers between 10 and 20 and identifying perfect squares between 800 and 900. The four colour theorem is discussed as a large-scale application of proof by exhaustion using computers. The tutorial then shifts to disproof by counterexample, illustrating how a single counterexample can invalidate a conjecture, with examples provided for clarity.

What is the main idea behind proof by exhaustion?

• To prove a statement by checking a few random cases.

• To prove a statement by checking all possible cases.

• To disprove a statement by finding a counterexample.

• To prove a statement using mathematical induction.

How many prime numbers are there between 10 and 20?

• Five

• Four

• Three

• Six

What is the significance of the four-color theorem in the context of proof by exhaustion?

• It shows that maps can be colored without any colors.

• It proves that all maps have the same number of regions.

• It demonstrates the use of computers in proving theorems by exhaustion.

• It shows that any map can be colored with three colors.

What is the key difference between proof by exhaustion and disproof by counterexample?

• Proof by exhaustion requires checking all cases, while disproof by counterexample requires finding one counterexample.

• Proof by exhaustion is faster than disproof by counterexample.

• Disproof by counterexample requires checking all cases, while proof by exhaustion requires finding one counterexample.

• Both methods require checking all possible cases.

Which of the following is a counterexample to the conjecture that x^2 is always greater than or equal to x?

• x = 1

• x = 2

• x = 3

• x = 0.5

For which value of n does the expression 2n^2 + 11 fail to be prime?

• n = 9

• n = 11

• n = 7

• n = 5

What is the result of expressing 253 as a product of two numbers?

• 21 x 12

• 11 x 23

• 13 x 19

• 17 x 15