Understanding Proof by Counter-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.

Conditional statements

Biconditional statements

Universal statements

Existential statements

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.

The quantifier remains the same.

The quantifier is removed.

The quantifier type changes.

The quantifier is doubled.

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.

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.

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.

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.

41 squared

A negative number

A composite number

A prime number

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.