If P is a set of integers and 3 is in P, is every positive
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28 Mar 2023, 23:19
Thank you all for the explanation, discussion and additional links to other similar questions. I finally got it. Thank you so much! I was stuck on understanding why statement (1) was sufficient. Let me try it in my own words:
Question stem says you have a set of integers that is called P. It further says that 3 is one of the integers in P. It asks if you can tell if all the positive multiples of 3 are in P.
Statement (1) says you can pick any integer from P and then add that integer to 3 and be sure that the result of this addition will also be included in the list.
You don't have to worry about all the possible values (e.g. 1, 2, 4, 5) in P, you don't have to try them out one by one to see if that addition will be one of 3's positive multiples. All you need to know is one thing and that is already given in the question stem -- that is 3 is in P and 3 is also an integer. So when Statement (1) says you can pick any integer from P, you can pick 3. Next, add that 3 from P to the 3 from Statement (1) to get 6. Then, as Statement (1) says the result of such addition will be included in P, you know 6 will be in P. Now 6 is in P, you can repeat the process by adding 3 to 6 to get 9. 9 will be included in the list. So the list will go on forever. You will get all positive multiples of 3 in P. Therefore, Statement (1) alone is sufficient.
You may question what if you don't pick 3 for Statement (1), will that make Statement (1) insufficient? This statement does not limit your selection of integer to 3 only, you may pick any other integer. However, you cannot exclude 3 from the list of "any integer" because the question stem already tells you that 3 is an integer listed in P. As long as you have 3 in P, you know Statement (1) alone is sufficient.
Statement (2) is similar to Statement (1) but it goes in a different direction. Statement (2) says you can pick any integer from P, then subtract 3 from your selected integer and be sure the result from such a subtraction will be included in P.
All you know about this question is that you have an integer "3" in P. That is the only thing you are sure. If you pick that 3 from P and minus the 3 from Statement (2), you will get 0 and 0 will be included in the P. Then, 0 minus 3 is negative 3. The list will go on. In this case, the only positive multiple of 3 in P is the original 3 from the question stem. Furthermore, since Statement (2) , Statement (1) and the question stem have not provided you with other information about the rest of the integers in P, you will have no clue whether all the positive multiples of 3 are in the list. Therefore, Statement (2) alone is not sufficient and Statement (1) & (2) together is not sufficient either.
The answer is A: Statement (1) alone is sufficient.
The remaining question is how am I supposed to do all this thinking in 2 minutes during the exam?