Roman Numeral Question on Set of Numbers

Hey

QPS00253 GMAT Prep Pack 1

"For a certain set of numbers, if x is in the set, then x-3 is also in the set. If the number 1 is in the set, which of the following must also be in the set?

I. 4

II. -1

III. -5

OA: C

OA explanation:

"The rule for set membership is: If x is in the set, then x-3 is also in the set. It is given that 1 is in the set. Therefore, 1-3 = -2 is in the set. Also, since -2 is now known to be in the set, -2-3 = -5 is also known to be in the set. Similarly, since -5 is now known to be in the set, -8 is also known to be in the set. Note that the set could be {1,-2,-5,-8,...} and this does not contain either 4 or -1. Therefore, it is not true that either 4 or -1 must be in the set. Only -5 must be in the set."

My question:

I initally set x=1 and consequently x-3= - 2 then -5 is in the set and so forth.
BUT, I also set x-3 = 1 through which I obtained 4 is in the set.

Am I right in assuming that we do not necessarily have to set x-3 = 1 to obtain 4 is in the set, because the question stem asked us

which of the following must also be in the set?

?

So setting x-3 = 1 is a "COULD BE TRUE" value but not a "MUST BE TRUE" value?

Appreciate your feedback..

Cheers, Tobi