Introduction to set theory
Learn to describe which values belong to a set, then use that description to count and select. Try the questions as you go.
Sets and number systems
Suppose the available seat numbers are 1, 3 and 5. We can collect them into a set, written $A=\{1,3,5\}$. The braces enclose the collection; $A$ is its name. Each object inside is called an element, or member. Asking whether seat 3 is available becomes a membership question.
$$3\in A\qquad 2\notin A$$
Read $\in$ as “is a member of” and $\notin$ as “is not a member of”.
We also need names for larger collections of numbers. A seat count uses whole numbers, a temperature can be negative, and half a metre needs a fraction. Each row below extends what the row above can describe.
| Set | What it contains | Examples |
|---|---|---|
| $\mathbb N$, natural numbers | Positive whole numbers used for counting. | $1,2,3,\ldots$ |
| $\mathbb Z$, integers | Whole numbers, including zero and negatives. | $\ldots,-2,-1,0,1,2,\ldots$ |
| $\mathbb Q$, rational numbers | Ratios $a/b$ of integers, with $b\ne0$. | $\frac12,\ -\frac34,\ 2=\frac21$ |
| $\mathbb R$, real numbers | All points on the number line, including irrational numbers that cannot be written as ratios of integers. | $\sqrt2,\ \pi,\ -1,\ \frac12$ |
| $\mathbb C$, complex numbers | Numbers $a+bi$, where $a,b$ are real and $i^2=-1$. | $2+i,\ -3i,\ 4=4+0i$ |
Where counting starts
Here $\mathbb N=\{1,2,3,\ldots\}$ and $\mathbb N_0=\{0,1,2,3,\ldots\}$. Some books include zero in $\mathbb N$. Check that convention when changing sources.
A number can belong to several of these sets. The integer $-1$ is rational because $-1=-1/1$, real because it lies on the number line, and complex because $-1=-1+0i$. The diagram places each example in its smallest set among these five. For complex numbers, recognizing $a+bi$ and the case $b=0$ is enough here; calculations with them come later.
When every member of one set belongs to another, the first is a subset of the second. We write $\mathbb N\subseteq\mathbb Z$. The symbol $\subsetneq$ adds that the larger set has extra members, so $\mathbb N\subsetneq\mathbb Z\subsetneq\mathbb Q\subsetneq\mathbb R\subsetneq\mathbb C$. Compare $3\in\mathbb N$, a number in a set, with $\{3\}\subseteq\mathbb N$, one set contained in another.
Classify the value after simplifying
- $\sqrt{81}=9$, so its smallest set in the chain is $\mathbb N$. A square-root sign alone does not make a number irrational.
- $0.125=125/1000=1/8$. This is rational but not an integer, so its smallest set is $\mathbb Q$. Terminating decimals, and decimals that eventually repeat, are rational.
- $\sqrt2$ is irrational, so its smallest set in this chain is $\mathbb R$. It still belongs to $\mathbb C$.
- $2+i$ has a nonzero imaginary part, so it belongs to $\mathbb C$ and not $\mathbb R$. By contrast, $2+0i=2$ belongs to $\mathbb N$.
When two sets are equal
Return to the available seats, $A=\{1,3,5\}$. Someone else writes $B=\{5,1,3,3\}$. These descriptions answer every membership question the same way. They therefore describe equal sets: $A=B$. The extra 3 does not create another seat.
To check equality, look for a member that occurs in just one set. For example, $\{1,3,5\}\ne\{1,3,7\}$ because 5 belongs only to the first. Both have three members, which is insufficient for equality.
The word “equivalent”
Some texts call equal sets “equivalent”; others use that word for sets whose members can be paired one to one, even when the members differ. We use “equal” for identical membership here. Equal sets always have the same members.
Read the braces one layer at a time
- $\emptyset=\{\}$ has no members. The set $\{0\}$ has one member, the number zero. They are different sets.
- $\{\emptyset\}$ also has one member, but that member is itself the empty set. An empty container and a container holding an empty container are different.
- In $\{1,\{1\}\}$, the number 1 and the set containing 1 are two different members. Removing repeated entries never means removing braces.
Different descriptions can have exactly the same members. The same principle will let us replace a long list with a short rule, provided the rule generates exactly the same members.