Understanding Proof by Counter-Example

The video tutorial explains the concept of proof by counter-example, emphasizing that it is not valid to prove a universal statement with just an example. It discusses existential statements, which can be proven with examples, and demonstrates this with the expression n squared minus n plus 41. The tutorial highlights the importance of negating statements correctly and provides a formal proof by counter-example, showing that the statement 'for all integers n, n squared minus n plus 41 is prime' is false by finding a counter-example.

Why is it generally not acceptable to prove a universal statement with just an example?

• Because examples only show a statement is true for specific cases.

• Because examples are too complex to understand.

• Because examples are only used for existential statements.

• Because examples are not valid in mathematics.

What type of statement can be proven by providing an example?

• Conditional statements

• Biconditional statements

• Universal statements

• Existential statements

What is the negation of the statement 'For all positive integers n, n squared minus n plus 41 is prime'?

• There is a positive integer n such that n squared minus n plus 41 is prime.

• For all integers n, n squared minus n plus 41 is not prime.

• There is a positive integer n such that n squared minus n plus 41 is not prime.

• There is no integer n such that n squared minus n plus 41 is not prime.

What happens to the quantifier when negating a statement involving quantifiers?

• The quantifier remains the same.

• The quantifier is removed.

• The quantifier type changes.

• The quantifier is doubled.

What is the logical equivalent of negating 'For every x, P(x) is true'?

• There exists an x such that P(x) is true.

• For every x, P(x) is false.

• There exists an x such that the negation of P(x) is true.

• For every x, the negation of P(x) is false.

Why is it important to be careful when negating statements involving quantifiers?

• Because it changes the meaning of the statement.

• Because it has no effect on the statement.

• Because it makes the statement longer.

• Because it simplifies the statement.

What is the purpose of finding a counter-example in a proof?

• To prove a statement is logically equivalent.

• To prove a statement is existentially true.

• To prove a statement is false.

• To prove a statement is universally true.

In the context of the video, what is a prime number?

• A number with only one factor.

• A number with exactly two factors: one and itself.

• A number with three factors.

• A number with no factors.

What is the result of n squared minus n plus 41 when n equals 41?

• 41 squared

• A negative number

• A composite number

• A prime number

What is the main goal of the proof by counter-example method?

• To prove a statement is true for all cases.

• To prove a statement is false by finding a single counter-example.

• To prove a statement is true for some cases.

• To prove a statement is logically equivalent.