If S is an infinite set of real numbers, is there a number

Question

If S is an infinite set of real numbers, is there a number in S that is less than every other number in S?

(1) Every number is S is an integer

(2) Every number in S is positive

Help to explain, thank you~~~

jamifahad · Answer

Are you sure OA is C?

Let S be set of finite number for simplification.

Stmt1: Every number is S is an integer
S={-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7}
is there a number in S that is less than every other number in S? yes. -5
S={1,1,1,1,1,1,1,1}
is there a number in S that is less than every other number in S? No.
Insufficient.

Stmt2: Every number in S is positive
S={1,2,3,4,5}
is there a number in S that is less than every other number in S? yes.1
S={1,1,1,1,1,1}
is there a number in S that is less than every other number in S? No.
Insufficient.

Together, S={1,2,3,4,5} 
is there a number in S that is less than every other number in S? yes.1
S={1,1,1,1,1,1,1}
is there a number in S that is less than every other number in S? No.
Insufficient.

OA E. Or am i missing something here?

iamyellow · Answer

same here. my answer is also E.

mehtakaustubh · Answer

agree with  jamifahad  it should be E

BenchPrepGURU · Answer

Yes - you are missing a couple somethings. 
(1) S is an infinite set. You can't simplify things and only consider finite sets. Consider the set of all integers: you can always find a smaller integer. Consider the set of all real numbers between 0 and 1 non-inclusive - whatever fraction you pick, I'll find a smaller one. Consider the set of real numbers between 0 and 1 inclusive - 0 is smaller than every number in that set.
(2) When the GMAT says set, it means collection of unique elements - S = {1, 1, 1, 1,} = {1}. Sets with repeated elements are not allowed.

With respect to (1) all positive integers - YES, all integers - NO
With respect to (2) 1 to 2 non-inclusive - NO, 1 to 2 inclusive - YES
Combined the two pieces of data are the well-ordering axiom which says that any non-empty set of natural numbers (positive integers) contains a least element.

So, the answer is C

jamifahad · Answer

True.
In that case, S will be Set of distinct positive integer S={1,2,3,4,5,6.......}
is there a number in S that is less than every other number in S? Yes. 1

Thanks. Learnt new thing today.

BenchPrepGURU · Answer

jamifahad - No problems. Infinite sets have many wierd and interesting properties. Search "Georg Cantor" and you'll find some stuff that will blow your mind.

MrMicrostrip · Answer

nice concepts....thanx for explaining....

abhi398 · Answer

explain me wat exactly the question wants us to find ??

manishgeorge · Answer

Bench Prep Guru, could you explain this a little better for my benefit.

I catch your statements

With respect to (1) all positive integers - YES ( are you considering zero as the smallest), all integers - NO (why no, cant there be a negative number which is smaller that all the other numbers)

Same confusion with the below one too
With respect to (2) 1 to 2 non-inclusive - NO, 1 to 2 inclusive - YES
Combined the two pieces of data are the well-ordering axiom which says that any non-empty set of natural numbers (positive integers) contains a least element.

BenchPrepGURU · Answer

With respect to (1) if we consider just the positive integers then that would be sufficient. 1 is the smallest positive integer (0 is neither negative or positive - if you want to include 0 you could use "non-negative integers"). When it comes to all integers there is no smallest number. We could play a game - you pick an negative integer, x, which you think is smaller than all other integers, and I'll pick x - 1. This game would never end because we're dealing with an infinite set.

With respect to (2) 
1 to 2 non inclusive means  1 < x < 2 
1 to 2 inclusive means  1 \leq x \leq 2 
Since we're considering real numbers and not integers here, there is no smallest number in the first set, while there is a smallest number in the second set - namely 1. This is a subtle difference, but an important one... the difference is subtle enough that it underlies an axiom about the real numbers that is fundamental to a rigorous development of calculus.

We can play a game here as well - 1.0001 is in the first set, and it might seem reasonable to claim that this is the smallest number in that set, but 1.00001 is also in that set. 1.(as many zeros as you want)1 is in the set as well, and we can always find a smaller number. Since we're dealing with the real numbers and not the integers, both of these sets are infinite.

We need positive and integers to be able to find a least element

manishgeorge · Answer

Understood... thanks for explaining this in great detail

petrifiedbutstanding · Answer

I didn't know this. Thanks. What other rules like this exist? Is there a book or a website you can point us to?

If S is an infinite set of real numbers, is there a number

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If S is an infinite set of real numbers, is there a number

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards