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I require some guidance with the following question:

Consider the following subsets of all integers. $$\begin{align*} A&=\{2n+1\mid n\text{ is an element of all integers}\}\\ B&=\{3n\mid n\text{ is an element of all integers}\}\\ C&=\{3n+2\mid n\text{ is an element of all integers}\} \end{align*}$$ Find each of the following sets, and express it in set-builder notation.

  1. $A-B$.
  2. $B\cap C$.
  3. $C\cap B^c$
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Please don't yell. (All caps are interpreted as yelling). – Arturo Magidin Mar 31 '11 at 20:00
This is the first time I see that terminology "set builder notation". – Adrián Barquero Mar 31 '11 at 20:03
@Adrian: Unfortunately common in (IMHO bad) books (of which there are far too many). – Arturo Magidin Mar 31 '11 at 20:05
@Ryan P: It's called the "complement of $B$", not the "inverse of $B$". – Arturo Magidin Mar 31 '11 at 20:07
Yes, I am sorry, I meant the complement of B. – user8958 Mar 31 '11 at 20:09

Can you describe in words what $A-B$, $B\cap C$, and $C\cap(B^c)$ are?

For example, for an integer to be in $B$, it must be a multiple of $3$. To be in $C$, it must be an even number plus $2$ (that is, it must be an even number). So to be in $B\cap C$, it must be both even and a multiple of $3$. Can you describe what numbers are both even and multiples of $3$? If so, then you can put that description into the "set-builder notation".

So, start by figuring out what is in each of the three sets (with words). We can then go on from there.

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@Arturo Magidin: Well, for the first question 'a', in words I see it as "All elements of sort Two times N plus one, which are not multiples of 3". – user8958 Mar 31 '11 at 20:16
@Ryan P: Can you say "of the sort two times N plus 1" with a single word, commonly used to describe certain kinds of integers? – Arturo Magidin Mar 31 '11 at 20:24
I guess it could be said that it is 'positive'. As we are adding 1 to each possible subset of A. – user8958 Mar 31 '11 at 20:30
@Ryan: No, it cannot be said "positive". Take $N=-50$. How much is "two times N plus one"? And no, we are not adding 1 to each possible subset of A, we are adding one to the result of multiplying an integer by 2. – Arturo Magidin Mar 31 '11 at 20:32
Hmm, well in any case 2n+1 would result in an odd integer. – user8958 Mar 31 '11 at 20:39

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