Question

What is a counterexample for the conjecture? Conjecture: The product of two positive numbers is greater than the sum of the two numbers.
A. 3 and 5

B. 2 and 2

C. A counterexample exists, but it is not shown above.

D. There is no counterexample. The conjecture is true.

Answer 1

Rewrite: The product of two positive numbers is greater than the sum of the two as an If, Then statement.

If you multiply two positive numbers, then the product of those numbers is greater than the sum of the two numbers.

Answer 2

A counterexample makes the "If" portion true but not the "Then" portion.

Consider the 3 and 5 of option A.

3*5 = 15. Then, according to the If-Then statement 15 must be > 3 + 5. 15 > 8.
Therefore, 3 and 5 are not a counterexample to the statement.

Answer 3

Next, follow that same line of thinking for the two numbers in option B.

Multiply 2*2 and then compare that product to (2 + 2).