$T$: a set of natural numbers.
$S_1$: $2$ is the only prime number that divides elements of $T$.
$S_2: T = \{16, 8, 528\}.$
I'm trying to figure out which statements imply each other, i.e., does $S_1$ imply $S_2$ and vice versa.
I'm not sure if $S_1 \implies S_2$, because $S_1$ doesn't seem to be necessary for $S_2$ to exist. But for $S_2 \implies S_1$, I think it is necessary that $2$ is the only prime that can divide $T$'s elements.
What do you guys think? Cheers in advance, much appreciated.
In order to show that $S_1$ does not imply $S_2$, you should give an example of a situation (that is, a set $T$) for which $S_1$ is true, but $S_2$ is false. Can you think of any?
Note that $S_2$ does not imply $S_1$ (assuming I am interpreting $S_1$ correctly). In particular, $528$ is divisible by $3$, which is a prime number not equal to $2$.