I am studying for the math subject GRE.
I have a book that gave the order "s is a multiple of t" and the set {2, 3, 4, 6, 8, 9}, and asked for the minimal elements.
The answer given is 6, 8, and 9.
I would have said 2 and 3, and everything I have found online seems to be in agreement.
Is the use of the word "multiple" what flips the direction?
If so, I would think that since saying something is multiple of something else is equivalent to using the word "divisor" with the "something"'s flipped and due to the arbitrariness of "s" and "t", using the word multiple should not be sufficient to flip the direction of minimality. And if so, can someone please explain how one is to know what it is explicitly about this wording that makes it that makes the direction certain?
I believe what your book is saying is that for $s, t \in \{2,3,4,6,8,9\}$, $s \leq t$ iff $s$ is a multiple of $t$. So for example, here $6<3$ since 6 is a multiple of 3. Thus since there are no multiples of 6, 8, or 9 in your set, there is nothing that is "less" than them, so they are minimal. On the other hand, $6<2$ since 6 is a multiple of 2, so 2 is not minimal, and so on.
Generally, orders are defined where the first variable given is less than the second based on whatever rule follows. So if a problem gives an order "$s$ is a multiple of $t$", then it means $s\leq t$ if $s$ is a multiple of $t$". If it gives an order "$z$ divides $k$", then it means $z\leq k$ if $z$ divides $k$.