K is a set of integers such that if the integer r is in K,

Question

K is a set of integers such that if the integer r is in K, then r + 1 is also in K. Is 100 in K?

(1) 50 is in K.
(2) 150 is in K.

Bunuel · Accepted Answer

(1) 50 is in K --> integers more than or equal to 50 are in the set K, so 100 is in the K. Sufficient. (2) 150 is in K --> integers more than or equal to 150 are in the set K, so 100 may or may not be in the K (if the source integer is 100 or less then 100 is in K but if the source integer is more than 100 then 100 is not in K). Not sufficient. Answer: A. Similar questions: https://gmatclub.com/forum/k-is-a-set-o ... 96907.html https://gmatclub.com/forum/for-a-certain ... 36580.html https://gmatclub.com/forum/if-p-is-a-set ... 96630.html https://gmatclub.com/forum/k-is-a-set-of ... 03005.html Hope it helps.

shrouded1 · Answer

If r is in the set, r+1 will be in it.
So if 50 is in the set, 51 will be in it
If 51 is in the set, by the same logic, 52 will be in it
If 52 is in the set, 53 will be in it .... AND SO ON

So basically what the condition implies is if r is in the set, all the integers greater than r will also have to be in the set. Hence, 50 being in there is sufficient for 100 to be in there. But 150 being in there, is not necessarily sufficient

muditadixit81 · Answer

questions such as the one above never mention the total number of terms in the set , wouldnt that matter , say if the set were to contain only 10 integers or say 50 , the answer to the question might change ....?

shrouded1 · Answer

Not really, because the way the set is defined, it will always be either the empty set or an infinite set.

If the set contains any integer, it will have to contain all the integers greater than or equal to that integer. So the question of having exactly 50 or 100 integers never arises

cano · Answer

Well, that is some valuable information. Thank you and +1.

vibhutirs · Answer

Guys, would you mind explaining the solution bit more. How can 100 be in set, if 50 is in K. Wouldn't r+1, i.e., 51 should be there? Thanks!

abhishekkpv · Answer

From 1
If 50 is in a part of K then 51 is a part of k and if 51 is a part of k then 52 is also a part of k
so the possible values of k are k>=50

From 2
k>=150

so from 1 we can say that 100 is a part of k but from 2 we are not sure if it is contains 100 or not...

Ans: A)

mumbijoh · Answer

Bunuel,
Hi I understood your explanation.However , I think if you were to classify statement 2 as insufficient then statement 1 would also be
if 50 is in set the 51,is also in the ,same as 52...... all the way to a hundred.
if 150 is in set then 149 must have been part of the set, 148 .. all the way to possibly 0.
But following your explanation you mentioned in statement two that there is no way of knowing if 100 was part of the set? Well i agree with you but how are we to know if statement 1 goes all the way to 100?
I was thinking that E would have been the answer.no?

Bunuel · Answer

We know that if r is in K, then r + 1 is also in K. (1) says that 50 is in K, thus every integer more than 50 is also in K: 51 because 50 is there, 52 because 51 is there, ..., 100 because 99 is there. Similar questions to practice: for-a-certain-set-of-numbers-if-x-is-in-the-set-then-both-161920.html for-a-certain-set-of-numbers-if-x-is-in-the-set-then-x-136580.html a-set-of-numbers-has-the-property-that-for-any-number-t-in-t-98829.html if-p-is-a-set-of-integers-and-3-is-in-p-is-every-positive-96630.html k-is-a-set-of-numbers-such-that-i-if-x-is-in-k-then-x-96907.html Hope this helps.

yenpham9 · Answer

Hi Bunuel,

Can you explain again why B is insufficient? If 150 is in K, 151, .152... resulting from ( r + 1) also is in K. That means 100 will be out of K because K includes only value > 150. B helps to always answer: NO (100 is always NOT in K: Sufficient).

Thanks.

Bunuel · Answer

I guess you did not follow the links provided...

We don't know which is the source integer in the set, if it's 150, then 100 won't be in the set but if the source integer is say 10 or 20 (basically if the source integer is less than or equal to 100), then 100 will be in the set. So, 100 may or may not be in the set. Consider below two examples of the set:

{150, 151, 152, ...}
{1, 2, 3, 4, ..., 100, ..., 150, ...}

Notice that both sets satisfy the second statement.

Hope it's clear.

gmatkum · Answer

If r is in K, r+1 is in K. So if 50 is in K, 51 is in K. But if 51 is in K, so is 52, and then so is 53, and so on. Indeed, if you know 50 is in K, you know that every positive integer larger than 50 must also be in K. So 1) is sufficient.

If 150 is in K, all we know for certain is that every positive integer greater than or equal to 150 must be in K. We don't know if 100 is in K. So 2) is insufficient.

Thus, A.

Coggi · Answer

Why is it not C ? 
We don't know how many integer there is in the set...
Maybe it's a set of 10 intergers, maybe a set of 500 ? Thus with 1) and 2) we know for sure 100 is in K 
What am I missing ?

freedom128 · Answer

Hi  Coggi ,

Option A alone is SUFFICIENT. If 50 is in the set, then 51, 52,..,100,...,to the infinity must be in the same set. 100% sure, and no additional data is even necessary.

Of course, option (C) must not be correct answer, because in Option (C), Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

One golden rule of thumb in DS is not to make any own assumption other than stated in the question stem. For this case, the question doesn't state nor explicitly indicate that there are finite number of the set.

Coggi · Answer

Okay I get it, so if no detail added always consider the set is infinite ?

freedom128 · Answer

Yes, it is natural to do that. You can try this better but similar PS/DS questions here: 1) https://gmatclub.com/forum/each-of-64-w ... 03791.html 2) https://gmatclub.com/forum/if-n-is-a-po ... 03255.html Posted from my mobile device

sohaib123 · Answer

this is badly worded question. If r is in K then r+1 is in k meaning 51 is in k. Why would 51 be considered r then. I thought the same thing earlier but there is no reason to consider the new 51 as r. It could be anything like f or how do we even know that this set is infinite. Not all information is given!

MHIKER · Answer

(1)  50;50+1=51...100,101  we can progress is in the way. Sufficient.

(2) If  r=99, then \ 99+1= 100 , and progressing this way  150   is in K, But if  r=149  then  149+1=150  is the k but  100   is not in the k. Yes and No; Insufficient.

The answer is  A

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K is a set of integers such that if the integer r is in K,

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