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$T$: 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.

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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 $2$.

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  • $\begingroup$ $528$ definitely is not a power of $2$. $\endgroup$
    – Lord_Farin
    Commented Jan 28, 2015 at 19:27
  • $\begingroup$ @Lord_Farin good catch. Fixed it now. $\endgroup$
    – Ben Grossmann
    Commented Jan 28, 2015 at 21:16
  • $\begingroup$ I didn't even notice that. Great input! Thanks @ Lord_Farin, Omnomnomnom $\endgroup$
    – Romaion
    Commented Jan 29, 2015 at 1:05

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