Describe the set of all odd numbers between $100$ and $200$ using set builder notation I've come across a question in Discrete Mathematics, asking me to use set builder notation to describe the set of all odd numbers between 100 and 200.
The answer I had was:
$$\{ p | p = 2n + 1, n \text{ (all numbers) } [50, 99], 100 < p < 200 \}$$
Although this should technically give the correct answer, the answers in the textbook have:
$$\{x\,|\,100<x<200\text{ and }2\not | x\}$$
I get the first part, however I have no clue what the end means (2 |/ x); what is that symbol called, and does that represent all odd numbers? 
 A: The symbol $\mid$ means 'divides'. Drawing a line through it to get $\not \mid$ means 'does not divide'. As for your answer, it is absolutely correct. There are many equally correct ways to write the set of odd numbers with setbuilder notation.
A: As others have noted, the $|$ symbol means "divide".  And when we put a slash through it, it means "does not divide".
What does it mean for $2$ to "divide" a number?  It means $2$ is a factor of that number.  For example, $2$ divides $6$ because $6 = 3 \cdot 2$ (so $2$ is a factor of $6$).  Similarly, $2$ divides $100$, since $100 = 5 \cdot 5 \cdot 2 \cdot 2$.  Meanwhile, $2$ does not divide $15$ since $15 = 3 \cdot 5$.
Similarly, $2$ divides every even number (isn't that how we define even numbers? as numbers having a factor of $2$?).
But if we define even numbers as numbers having a factor of $2$ in them, then odd numbers are numbers without a factor of $2$ in them.  That means $2$ is not a factor of any odd number, which means $2$ does not "divide" any odd number.
So, when we say that $100 < x < 200$ and $2 \not | x$ (i.e., $2$ does not divide $x$), we are saying $x$ is between 100 and 200, and $2$ does not divide $x$, i.e., $x$ is not even, i.e., $x$ is odd.
By the way, here is an extra question for you.  Could the answer also be $\{ x | 100 \leq x \leq 200 \text{ and } 2 \not | x \}$?  Why or why not?
