Claim: if two integers have product 0, then both integers equal 0. Which pair refutes it?

• a = 0, b = 0
• a = 2, b = 3
• a = 0, b = 7
• a = 3, b = 4

Claim: all prime numbers are odd. Which counterexample breaks it?

• 9
• 2
• 15

What is the negation of the claim all A are B?

• Some A is not B
• No A is B
• All B are A

Testing a conjecture at n equals 0 and n equals 1 is a waste of time; only large random cases matter.

Argument: 'If P then Q; Q is true; therefore P is true.' Which assignment makes both premises true and the conclusion false?

• P is false; Q is true
• P is true; Q is true
• P is true; Q is false
• P is false; Q is false

Several differentiable functions have been tested, and each is continuous at c. What do those tests alone establish about the claim that differentiability at c implies continuity there?

• Every continuous real-valued function must be differentiable at that same point.
• Every differentiable real-valued function has now been checked, so no proof is needed.
• The tested cases fit; a general proof is still needed.

Claim: the product of any two irrational numbers is irrational. Which pair breaks it?

• Root 2 times root 3
• Root 5 times root 7
• Root 2 times root 2

Claim: x squared is positive for every nonzero real x. Sam proposes x=0, where x squared is 0. Why is this not a counterexample?

• It satisfies the hypothesis but has a positive square.
• It does not satisfy the stated nonzero hypothesis.
• It satisfies the hypothesis and refutes the conclusion.
• A counterexample must always be a negative real number.

Let f and g be real-valued functions defined near c. A discontinuous pair with g=-f refutes the claim that two discontinuous functions must have a discontinuous sum at c. Which replacement is true?

• If f is continuous at c and g is discontinuous there, their sum is discontinuous at c.
• If f and g are both discontinuous at c, their sum must be discontinuous at c even when g=-f.
• If f and g are both continuous at c, their sum can be discontinuous at c despite the continuity of both.

The claim 'every continuous real-valued function on a nonempty interval attains a maximum' fails for f(x)=x on (0,1). Which restriction makes the claim true?

• The domain is bounded but may leave its endpoints out.
• The domain is closed but may extend without any bound.
• The function is differentiable on its open interval.
• The domain is a nonempty closed bounded interval.