Conjectures and Counterexamples

Which of the following is a counterexample to the following conjecture? If  $x^2\ =\ 4$ , then x = 2

• x = 4

• x = -2

• x = 2

• x = -4

A concluding statement reached using inductive reasoning is called a _______

• compound statement

• conjecture

• condition

• counterexample

Which answer would be a counterexample to the following biconditional?

"Greg is a swimmer if and only if he is an athlete."

• Greg is an athlete and a basketball player.

• Greg is a swimmer.

• Greg is not an athlete

• There is no counterexample.

A counterexample is an example that proves a conjecture to be true.

• True

• False

For any right triangle, the sum of the squares of the legs is equal to the square of the hypothesis. This is an example of a conjecture.

• True

• False

Conjectures, like theorems, have been proven to be true.

• True

• False

No number of examples or cases can fully prove a conjecture.

• True

• False

Which number is a counterexample to the following statement?
For all numbers a, 2a + 7 $\le17$  

• a = 6

• a = 0

• a = 5

• a = 1

Which numbers are not counterexamples for the following statement?
For any numbers a and b,  $\frac{a}{b}\ =\ a\ -\ b$  

• a = 8, b = 4

• a = 10, b = 5

• a = 6, b = 3

• a = 4, b = 2

To fully disprove a conjecture, one needs to find only ONE counterexample.

• True

• False