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Please help me out on this question.

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Please help me out on this question.

by aman88 » Sat Dec 01, 2012 11:20 am
This question,

K is a set of integers such that if the integer r is in the K, then r+1 is also in K. Is 100 in K?
1. 50 is in K
2. 150 is in K

I got to know (from previous discussions) that the answer is A, but my question is:

Isn't it important to consider how many integers are there in the set K? Say, if we have 10 integers in set K, then there will not be a 100 in it. And if we have 50 or more, then there will be 100. So ambiguous! How did we reach to a conclusion that there will be infinite integers?

Thanks.
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by Anindya Madhudor » Sat Dec 01, 2012 11:34 am
Statement I:
If 50 is in the set, then 51 will also be in the set, 52 will be in K, etc. So, any integer greater than 50 will be in the set. Which means 100 will definitely be in the set. Sufficient.

Statement II:
Anything greater than 150 will be in K. But, we do not know what is the first term in the set. If it is less than 100, then 100 is in K. But, if the first term is greater than 100 then 100 will not be in the set. Insufficient.
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by aman88 » Sun Dec 02, 2012 2:21 am
Thanks but isn't it important to consider how many integers are there in set K? Say, if we have 10 integers in set K, then there will not be a 100 in it. And if we have 50 or more, then there will be 100. So ambiguous! How did we reach to a conclusion that there will be infinite integers?

Thanks.
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by Brent@GMATPrepNow » Sun Dec 02, 2012 8:11 am
aman88 wrote:Thanks but isn't it important to consider how many integers are there in set K? Say, if we have 10 integers in set K, then there will not be a 100 in it. And if we have 50 or more, then there will be 100. So ambiguous! How did we reach to a conclusion that there will be infinite integers?

Thanks.
Set K must contain an infinite number of values. We know this because there cannot be a greatest integer in set K.

For a moment, let's pretend that there is a greatest integer in set K, and then we'll demonstrate that this scenario would be impossible.

Aside: this method of reasoning is called "Proof by contradiction" (see: https://en.wikipedia.org/wiki/Proof_by_contradiction for more info)

Okay, let's say that N is the greatest integer in set K.
The question tells us that if integer r is in the K, then r+1 is also in K.
So, if N is in set K, then N+1 must be in set K.
If N and N+1 are both in set K, then N cannot be the greatest integer in set K.
But, we began by defining N as the greatest integer in set K.
This contradiction tells us that there cannot be a greatest integer in set K, which means set K contain an infinite number of integers.

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by aman88 » Sun Dec 02, 2012 9:54 am
Thanks a lot Brent. That explanation cleared the very particle of doubt from my brain.

Thanks a ton!
Aman.
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