<p>If you can get through AP Calc without any trouble, you’re fine.</p>
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<p>Only if you explicitly define ‘infinite’, however if you just merely reason it out, by exclusion, all elements of N is taken out from Z, and there are still elements left over in Z, then Z is clearly larger than N.</p>
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<p>Took me a couple of tries to realize that it’s pronounced monotoni-city</p>
<p>Both answers make sense to me.</p>
<p>larger is a vague term</p>
<p>That’s true. Because isn’t infinity times two = infinity? I know infinity isn’t a number, but I wonder.</p>
<p>From the original question
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<p>The thing is, you can count the natural numbers 1,…, infinity. Then you can count the integers, starting at 0 and switching from positive to negative…assigning a “count” to each integer, you can count integers 1,…,infinity. By this logic they have equal number of elements.</p>
<p>However, you can also take all of the natural numbers, count them up, multiply by 2 to include negative integers, and add 1 for 0. Then you have 2N+1=I.</p>
<p>So the question is, is 2*infinity+1> or = infinity? Generally they are called equal. Of course, this makes no sense intuitively, but there you have it.</p>
<p>Man In The Mirror :)</p>
<p>Infinity is an absolute value, it cannot be used as a coefficient to measure the relationship between two sets/variables as we are doing right now.</p>
<p>Let |N| = lim -> oo, and |Z| = 2|N|+1;
|Z|/|N| = 2(|N|)/(|N|) + (lim->oo)^-1, which gives us 2. Given that |Z| is twice that of |N|, we have to say that Z is the larger set.</p>
<p>I’m in the same boat as you, OP. SAT math is really common sense in my opinion. I always score close to 800, but I always get confused in class or online when I encounter a difficult problem that I haven’t learned how to do. So no, we’re probably not good at math. However, I think that we can blame the education system! I hated math until this year, but I never knew I wasn’t good at math because I always got high 90’s in class. So really the only solution is to ponder problems like these and practice problem solving skills at a level higher than the common sense problem solving on the SAT. If you like math you should try to get better at it though! (that’s what I’m doing :))</p>
<p>For what it’s worth, |N| = |Z|</p>
<p>^Yes, b/c of infinity…I just don’t like it intuitively.</p>
<p>If |N| = |Z|, then the system that encloses both set N and set Z is not a rational system. Remember, absolute infinity can never exist in a rational system as it presumes that the boundaries of a rational system converges towards an absolute. Therefore |N| cannot equal |Z|</p>
<p>I’m not going to disagree with you, except that</p>
<p>|N| = |Z| < |R|</p>
<p>^And this is b/c R can’t be counted like integers?</p>
<p>Weird, I understand this discussion [/pride]</p>
<p>Right, integers and natural numbers are countably infinite. Real numbers are uncountably infinite.</p>
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<p>I am surprised that you did not try to use a similar technique to answer this question as you used to compute the sum of the integers from 1 to n inclusive.</p>
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The two problems seem to be of two distinct types, if we were to use the same methodology then we would justify that because the sum of set Z is less than the sum of N, N is larger than Z, whereas I assumed that the justification stems from the number of elements within the set.</p>
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<p>That I agree with.</p>
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<p>Do all infinite sets have the same size? For example, which set is larger, the natural numbers or the set of subsets of all of the natural numbers ({{}, {1}, {2}, {1,2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, {4}, …})?</p>
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<p>This might be a good jumping-off point to talk about what it means for something to make sense intuitively, as part of a larger discussion about what it means for someone to be “good at math”. For example, I might have posed my questions somewhat differently, in which case more people might have sooner figured out what I was trying to get them to think about.</p>
<p>Which set is larger, the even integers or the integers?</p>
<p>If someone has been given a simple definition of infinity, some of what’s been discussed here might not make sense intuitively. OTOH, if they’ve been encouraged to explore the relations between sets, it might.</p>
<p>At any rate, my goal here is to provide a framework for discussion about acquiring mathematical skills. When talking about people who do mathematics for a living, not only do they answer difficult questions, they also ask them. They take a problem that is known and modify it in other ways, e.g. after determining the size of a set they ask questions about subsets of that set, and so forth.</p>
<p>FWIW, there are plenty of web sites that give answers to these and related questions, if you do a little googling or hunting around wikipedia.</p>