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The number of subsets {1,2,…,n} with odd cardinality is ?

(Is it asking about number of subsets, we have to take power set then?)

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marked as duplicate by JMoravitz, kingW3, Xander Henderson, SchrodingersCat, Ben Sheller Oct 6 '17 at 18:02

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  • $\begingroup$ Hint: The number of subsets of $\{1,2,\dots,n\}$ with cardinality $r$ is $\binom{n}{r}=\frac{n!}{r!(n-r)!}$, the binomial coefficient "$n$ choose $r$." $\endgroup$ – JMoravitz Oct 5 '17 at 18:57
  • $\begingroup$ Let A={1,2} , P(A)={{},{1},{2},{1,2}}.So,odd cardinality= 3 ? $\endgroup$ – user3337199 Oct 5 '17 at 18:58
  • $\begingroup$ What three things do you count there? Remember that zero is an even number. $\endgroup$ – JMoravitz Oct 5 '17 at 18:59
  • $\begingroup$ ok got it. It should be 2 $\endgroup$ – user3337199 Oct 6 '17 at 4:54
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For $n>0$ There are half subset containing an odd and half containing an even number of elements

The set of subsets has $2^n$ elements so the number of subsets with odd cardinality

is half $2^n$ that is $2^{n-1}$

Hope this helps

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  • $\begingroup$ This answer doesn't give any intuition whatsoever as to why this might be the case. $\endgroup$ – JMoravitz Oct 6 '17 at 11:57
  • $\begingroup$ @JMoravitz There are half containing an odd and half containing an even number of elements. It is almost obvious. You are mean to downvote my beautiful answer now that I have edited and explained please rethink to your decision. $\endgroup$ – Raffaele Oct 6 '17 at 12:03
  • $\begingroup$ I disagree that it is "obvious." Although it is true that half contain an odd number of elements, a claim like that requires proof. Without proof this is as valid as saying "half of all milk drunk in the united states is chocolate milk." It could be true that half of all milk consumed actually is chocolate milk, but there is no reason to believe that ahead of time without evidence. $\endgroup$ – JMoravitz Oct 6 '17 at 12:08
  • $\begingroup$ @JMoravitz Are you triggered because you discovered this :)) Looks like everybody but you find it obvious. math.stackexchange.com/questions/267755/… $\endgroup$ – Raffaele Oct 6 '17 at 12:13
  • $\begingroup$ Whether I find it obvious or not is besides the point (I understand the proof and how to prove it, unlike you it seems). What is important is being able to convey the ideas of the proof to the student trying to learn. It helps noone to make unjustified claims and actively hurts in the case that your unjustified claims turn out to be flat out incorrect and false despite being "obvious." Including a proof or at least important observations that directly lead to a proof will help protect against false information and will help the student develop the intuition and skills to prove in future. $\endgroup$ – JMoravitz Oct 6 '17 at 12:18

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