---
title: "Introduction to set theory"
description: "Expand a constructive set definition into roster form: apply the rule to every allowed index, drop repeated outputs, and read the ellipsis correctly."
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md_url: "https://fanout.sh/ml-math/lessons/sets-building-sets-by-listing-members-t-0z1zgrp.md"
access: "public"
---

# Introduction to set theory

Expand a constructive set definition into roster form: apply the rule to every allowed index, drop repeated outputs, and read the ellipsis correctly.

Learn to describe which values belong to a set, then use that description to count and select. Try the questions as you go.

## Constructing a set

We will use $S=\{-2,-1,0,1,2\}$ throughout the next sections. Listing the members inside braces is called roster notation. A roster is easy to inspect, but a rule is more useful when a collection is large or infinite.

Here is a rule that makes $S$: take each of the inputs $0,1,2,3,4$ and subtract 2. The letter $n$ stands for whichever input we are using. The expression $n-2$ produces an output.

| Input $n$ | Calculation $n-2$ | Output |
| --- | --- | --- |
| 0 | $0-2$ | $-2$ |
| 1 | $1-2$ | $-1$ |
| 2 | $2-2$ | 0 |
| 3 | $3-2$ | 1 |
| 4 | $4-2$ | 2 |

$$S=\{n-2:n\in\{0,1,2,3,4\}\}$$

Read: the set of outputs n minus 2, where n runs through the listed inputs.

This is a constructive definition: the expression before the colon constructs members from the allowed inputs after it. Those inputs form the indexing set. Replacing $n$ with $k$ everywhere leaves the set unchanged. Changing the allowed inputs or the expression can change its members.

### Different inputs can produce the same member

1. Choose inputs $n\in\{-2,-1,0,1,2\}$ and the rule $n^2$, meaning n multiplied by itself.
2. The outputs are $4,1,0,1,4$. As a set, they become $\{0,1,4\}$.
3. There were five inputs but only three distinct outputs. The equality rule from the previous section removes the repetitions.

An infinite input set works too: $\{2n:n\in\mathbb N\}=\{2,4,6,\ldots\}$. Here the dots mean “continue doubling each positive whole number”. Stating the rule removes the guesswork that an unfinished list could leave.

## Defining a set by a condition

There is another way to describe $S$. Start with the integers and keep those at least $-2$ but less than 3. This time we select candidates directly, rather than transform inputs into new values.

$$S=\{x\in\mathbb Z:-2\le x<3\}$$

Read: the set of integers x such that minus 2 is less than or equal to x, and x is less than 3.

The candidate domain is $\mathbb Z$. The condition $-2\le x<3$ makes two demands at once: $x\ge-2$ and $x<3$. A candidate must satisfy both. This condition is called a predicate: supplying a particular number gives a statement that is true or false. The resulting set collects the candidates for which that statement is true.

### Check the endpoints and the domain

1. At $x=-2$, both $-2\le-2$ and $-2<3$ are true, so $-2$ is included.
2. At $x=3$, the test $3<3$ is false. The upper endpoint is excluded.
3. The fraction $1/2$ satisfies both inequalities, but it is not an integer. It fails the domain requirement.
4. The accepted integers are exactly $-2,-1,0,1,2$, recovering the roster for $S$.

The condition has a truth value once a candidate is supplied. The set itself does not: $S$ is a collection, while $2\in S$ is a true statement about that collection.

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