# 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

## 2 thoughts on “Homework Hints: Set-theoretic notation”

1. Fuzail Khan says:

Dear Prof. Reitz,

Thank you for the examples. It helped me for the homework. But, I have confused about the question: {x is an element of “Z” : |2x| <5}. Can you please explain me little bit about this question?

Sincerely,

Fuzail

1. Jonas Reitz says:

Hi Fuzail,

This one looks a little strange, because we are expecting a formula to come first, and instead we have $x\in \mathbb{Z}$. This is something common in mathematics that doesn’t exactly follow the rule, in which we describe the type of number (integers) in the first part of the set-theoretic notation – really, it means the same thing as this: $\{ x : x\in \mathbb{Z} and |2x|<5 \}$.

I hope this helps – feel free to write back if not.
-Prof. Reitz