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 October 21st, 2011, 01:08 AM #1 Newbie   Joined: Oct 2011 Posts: 16 Thanks: 0 Help with notation I've never seen Hi all, I was wondering whether you could help with an issue I have. Basically I have no idea what any of this means. I was given these exercises and I don't even know where to start. So is $\underline{j}$ the set of {0,1,2,...,j-1}? So I know that j and p are elements of the natural numbers (a) so does this mean that the cardinality of set j is equal to some element (b) I have no idea here? what is it asking (c) again no idea So my questions, What does the underline mean? And can someone help me work through each of the exercises above? As you have probably worked out I am new to discrete maths and set theory so any help would be much obliged. Thanks Dina
 October 21st, 2011, 05:15 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Help with notation I've never seen a asks if the cardinality of {0, 1, 2, ..., j-1} is j. b asks if the cardinality of {0} plus the cardinality of {0} is the cardinality of {0, 1}. etc.
 October 21st, 2011, 05:23 AM #3 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Help with notation I've never seen b) doesn't appear to be about the cardinality of the sets; I am unsure as to whether S+T in this context refers to union of S and T or the set $\{s+t\,:\,s\in S,\ t\in T\}$ (although in this case it's the same thing). $S\times T$ is the set $\{(s,t)\,:\,s\in S,\ t\in T\}.$
 October 21st, 2011, 05:25 AM #4 Newbie   Joined: Oct 2011 Posts: 16 Thanks: 0 Re: Help with notation I've never seen Hi thanks, so is this right? (a) True - cardinality of $\underline{j}$ is going to be a natural number (b) True - Cardinality of $\underline{1}$ + $\underline{1}$ is the cardinality of $\underline{2}$ (c) Not to sure on this one. How would I show this? I get the impression it's false, but that's just from intuition.
October 21st, 2011, 05:38 AM   #5
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Re: Help with notation I've never seen

Sorry -- as mattpi points out, b doesn't use cardinality symbols. I'm not sure what + is supposed to mean; what does your book have for it?

Quote:
 Originally Posted by Dina Soto (a) True - cardinality of $\underline{j}$ is going to be a natural number
That's not what it's saying. Re-read the question.

For c, do you understand what it is saying?

 October 21st, 2011, 06:19 AM #6 Newbie   Joined: Oct 2011 Posts: 16 Thanks: 0 Re: Help with notation I've never seen My book doesn't state anything for '+' or at least not what I can see. I am now really confused. So (a) isn't true? surely the cardinality of $\underline{j}$ is the natural number j, or does it not mean this. And for (c) I have no idea what it means. Perhaps give me an example. This is really confusing me, thanks for all the help and your time and trouble Dina
October 21st, 2011, 06:29 AM   #7
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Re: Help with notation I've never seen

Quote:
 Originally Posted by Dina Soto So (a) isn't true?
I didn't say whether it was true or false. But just knowing that the cardinality of the set is a natural number is not enough to determine whether the answer is true or false.

Once you understand a, c becomes easy.

 October 21st, 2011, 06:38 AM #8 Newbie   Joined: Oct 2011 Posts: 16 Thanks: 0 Re: Help with notation I've never seen Ok I really don't understand (a). Surely the cardinality of any set is a natural number. Lets assume that we're including 0 in the set of natural numbers, in which case surely (a) is always true? Could you explain (a) in more detail to me? I'm sorry for being such an idiot but this is really confusing me. Thanks Dina
October 21st, 2011, 07:02 AM   #9
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Re: Help with notation I've never seen

Quote:
 Originally Posted by Dina Soto Ok I really don't understand (a). Surely the cardinality of any set is a natural number.
First of all, that's false. But let's ignore that for a moment -- the cardinality of all the sets you're asking about are natural numbers.

But that's not what the problem asks! It's asking if $\forall m:\ |\underline{m}|=m$, not if $\forall m:\ \exists n:\ |\underline{m}|=n.$ Do you see the difference?

 October 21st, 2011, 07:14 AM #10 Newbie   Joined: Oct 2011 Posts: 16 Thanks: 0 Re: Help with notation I've never seen Ok I do see the difference, in that case (a) is true since $|\underline{j}|$ is equal to j, if j is 1 for example then $|\underline{j}|= \{0\}$ and it's cardinality is 1, not just any arbitrary natural number. Is that correct? Thanks Dina

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