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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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    Yelling has been fixed. :)2011-03-31

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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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    @Ryan P: I have to go, so I can't afford the time to keep pulling teeth: If you divide an integer by 6, the remainders can be $0$, $1$, $2$, $3$, $4$, of $5$. If you divide an *odd* integer by $6$, the remainder will have to be either $1$, $3$, or $5$. But if the number leaves a remainder of $3$ when divided by $6$, then it can be written as $6k+3 = 3(2k+1)$, so it is a multiple of $3$. So an odd number that is *not* a multiple of $6$ will necessarily leave a remainder of either $1$ or $5$, so it can be written as a multiple of $6$ plus or minus $1$, $6k\pm 1$.2011-03-31