MHB How to show uniqueness in this statement for integers

[cbarker1]

Dear Everyone,

Directions: Decide whether the statement is a theorem. If it is a theorem, prove it. if not, give a counterexample.

There exists a unique integer n such that $$n^2+2=3$$.

Proof:
Let n be the integer.

$$n^2+2=3$$
$$n^2=1$$
$$n=\pm1$$

How show this is unique or not? Please explain why if not.

 

[I like Serena]

Dear Everyone,

Directions: Decide whether the statement is a theorem. If it is a theorem, prove it. if not, give a counterexample.

There exists a unique integer n such that $$n^2+2=3$$.

Proof:
Let n be the integer.

$$n^2+2=3$$
$$n^2=1$$
$$n=\pm1$$

How show this is unique or not? Please explain why if not.

Hi Cbarker1,

You found 2 integer solutions that indeed satisfy the equation.
Doesn't that mean that the solution is not unique?
Our counter example is the fact that both n=-1 and n=+1 are solutions.
It would be different if we were only looking at natural numbers, excluding negative numbers, but 'just' integers can be negative.

 

[cbarker1]

Hi Cbarker1,

You found 2 integer solutions that indeed satisfy the equation.
Doesn't that mean that the solution is not unique?
Our counter example is the fact that both n=-1 and n=+1 are solutions.
It would be different if we were only looking at natural numbers, excluding negative numbers, but 'just' integers can be negative.

Yes, since I found two solutions to satisfy the equation. therefore, n is not unique.