# 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?

• It seems to be $2$ does not divide. $x$ Commented Mar 9, 2015 at 3:34
• Usually the diagonal strike would go through the divides by vertical strike, so you would get $2 \not | x$ to show that $2$ does not divide $x$ The use of $[50,99]$ and $100 \lt p \lt 200$ is redundant, but still correct. Commented Mar 9, 2015 at 3:36
• Don't use p. It usually means p is prime. Commented Mar 9, 2015 at 3:38
• It'd likely be better to either not specify the range in which $n$ is in as $$\{p | p=2n+1,n\text{ is an integer},100<p<200\}$$ - since that bit is redundant. You could also not specify where $p$ lies - and even get rid of $p$ altogether - to get $$\{2n+1 | 50\leq n \leq 99\}.$$ (Your answer is correct, but it's not in its simplest form) Commented Mar 9, 2015 at 3:43
• Thanks for your help guys. Some helpful tips and information. Commented Mar 9, 2015 at 3:44

## 2 Answers

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.

• So 2 does not divide by x is another way of writing all odd numbers? Commented Mar 9, 2015 at 3:37
• That is correct. The odd numbers are precisely those not divisible by $2$. Commented Mar 9, 2015 at 3:39

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?

• Very nice detailed answer. Thanks a lot! To your extra question: although it could be written as <=, its technically not correct because both 100 and 200 are even numbers, meaning that they will never be reached. (Is this correct? :P) Commented Mar 9, 2015 at 3:52
• @zuc0001 Well, I think you have the right idea. It is still correct to write the set with $\leq$, because the $100$ and $200$ might satisfy first condition, but as you suggested, they don't satisfy the second condition since both are even. So they wouldn't be in the set anyway. Commented Mar 9, 2015 at 3:54