# Homework Hints: Set-theoretic notation

Hi everyone,

I wanted to give an example of writing a given set in set-theoretic notation – this should help out with some of the problems on your first homework assignment (Section 1.1).

Example: Write in set-theoretic notation:  $\{5,10,15,20,\ldots \}$

In this case, you can see that the given set consists of all multiples of 5.  A good way to approach problems like this is to start with one of the basic sets, for example

•  the natural numbers $\mathbb{N} = \{1,2,3,4, \ldots \}$
• the integers $\mathbb{Z} = \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$

In this example, I can see that multiplying every natural number by 5 should give me the set that I want.  Therefore, I will use the formula $5n$, and the condition $n\in \mathbb{N}$.  Combining these in set-theoretic notation gives the solution:

Solution $\{ 5n : n\in \mathbb{N} \}$

Here are two ways to read this solution aloud:

• “the set of all $5n$ such that $n$ is a natural number”, or
• “for each $n$ in the natural numbers, multiply $n$ by $5$ and include the result in the set”

I hope this helps – feel free to respond here if you any questions.

Best,
Prof. Reitz