There is some integer $n$ such that if $n > 2,$ then $n^2 = 2n.$

Question

At the first glance, I think that

There is some integer $n$ such that if $n > 2,$ then $n^2 = 2n.$

is false. Even with careful contemplation, I still think that it is false.

However, I also agree with the answer key that the above statement is vacuously true, because $$2>2 \rightarrow 2^2 = 2(2)$$ is true, because $2>2$ is false.

I do not know how to resolve this. Anyone has a way to explain it?

Answer 1

If $A$ is false then $A \to B$ is always true, no matter what $B$ is. So just take an $n$ for which the first clause $n > 2$ is false, and then "if $n > 2$ then $B$" is true.

Similarly "if I am a unicorn then pigs can fly" is a true statement, since I am in fact not a unicorn. And "There is some person $A$ such that if $A$ is a unicorn then pigs can fly" is true as long as there is some person who is not a unicorn.

I suspect your confusion may be caused by the given statement looking almost like "There is some $n > 2$ such that $n^2 = 2 n$": in fact students often write "There is some $n$ such that if $n > 2$ then $\ldots$ when they mean ""There is some $n > 2$ such that $\ldots$". But the two statements are quite different.

Answer 2

The basic principle is that, for any logical propositions $A$ and $B$ we have:

$~~~~~~A \implies [\neg A \implies B]$

Regardless of the truth value of $B$, if $A$ is true, then the implication $\neg A$ implies $B$ is said to be vacuously true.

Here is the truth table (screenshot from Wolfram Alpha):

Here is a proof using a form of natural deduction (screenshot from DC Proof 2.0):

Answer 3

Expanding on Robert's observation:

1. This is the given sentence, which is true:

   • there is some integer $n$ such that if $n > 2,$ then $n^2 = 2n$
   • $\exists n{\in}\mathbb Z \:\big(n > 2\:\to\: n^2 = 2n\big).$

2. This sounds like the given sentence, but it is false:

   • there is some integer $n> 2$ such that $n^2 = 2n$
   • $\exists n{\in}\mathbb Z \:\big(n > 2\:\land\: n^2 = 2n\big).$