Confusion about the universal statements in determining whether it is True or False

Question

$1)$ $n=2$ only if $n^2-n-2=0$

$2)$ $n=2$ if $n^2-n-2=0$

$3)$ $n=2$ is sufficient for $n^2-n-2=0$

$4)$ $n=2$ is necessary for $n^2-n-2=0$

As we plug in $n=2$, all the above statements satisfy the given condition. But, how could I determine which of the above statements are True or False?

Answer 1

The whole point of this exercise is to make you read the words that surround the algebra.

In (1) the "only if" asks if you must have $n^2 - n - 2 = 0$ when $n=2$.

In (2) the "if" asks whether knowing $n^2 - n - 2 = 0$ guarantees $n=2$. Does that quadratic have any other roots?

In (3) the "is sufficient" asks whether knowing $n=2$ is enough to guarantee $n^2 - n - 2 = 0$ .

In (4) the "is necessary" echoes the question in (1).

If you're going to do well in discrete mathematics you have to practice paying attention to the logical connectives. They are (in a sense) more important than the (often easy) algebra.

Answer 2

Take the first statement. The question is: does $n=2$ only if $n^2-n-2=0$? This is the same thing as asking whether $n=2\implies n^2-n-2=0$. Yes, and you know it.

Now, take the second statment. Now, the question asks whether $n^2-n-2=0\implies n=2$. Now the answer is negative, since $n$ could also be equal to $-1$. Can you do the other two now?