Disprove the statement by giving a counterexample.

Question

I am confused about this question. It says

Disprove the statement by giving a counterexample.

For all positive integers $n$, $n^2- 2n$ is positive.

Then it gives me these choices:

4

1

0

1/2

2

So I did my math and pugged in the first one (4)

And got $4^2 - 2(4) = 16 - 8 = 8$

So would I check mark $4$ as a answer, or do I need a answer where the final result is NOT positive, like plugging in $1$, which will give me a answer of $-1$ ?

Answer 1

A counterexample means an example where the condition is true, but the conclusion is false.

• The condition is usually the part of the statement that comes after "if..." or "for all..." Here, the condition is "$n$ is a positive integer" because the statement says "for all positive integers $n$, ...".
• The conclusion is the other part of the statement. Here, the conclusion would be "$n^2-2n$ is a positive integer."

Therefore, $n=1$ or $n=2$ would be valid counterexamples because they are positive integers AND $n^2-2n$ is not positive. Both the condition is true and the conclusion is false.

However, $n=0$ and $n=1/2$ would not be valid counterexamples because the condition is not true: They are not positive integers.

Answer 2

You want to disprove the statement, so you want to produce an answer that makes the statement "$n^2 -2n$ is positive" incorrect. A counterexample would be an integer value of $n$ which shows this to be false; choose $n = 1$.