Sets and Set-Builder Notation

A set is the simplest kind of mathematical object: a collection, with nothing but its members. You already describe collections with predicates and domains; set notation packages that idea so that collections can be named, compared and counted.

Key idea

A set is determined by its elements; order and repetition do not matter, so $\{1, 2\} = \{2, 1\} = \{1, 1, 2\}$. Write $x \in A$ when $x$ is an element of $A$. $A \subseteq B$ ($A$ is a subset of $B$) means every element of $A$ is in $B$. $|A|$ is the number of elements of a finite set, and $\varnothing$ is the empty set, with $|\varnothing| = 0$.

A set can be written in roster form, listing its elements, as in $\{2, 3, 5, 7\}$, or in set-builder form, $\{x \in D : P(x)\}$, "the $x$ in $D$ for which $P(x)$ is true." Standard domains: $\mathbb{Z}$ (integers), $\mathbb{Z}^{+}$ (positive integers), $\mathbb{Q}$ (rational numbers), $\mathbb{R}$ (real numbers). Books disagree about whether $0$ belongs to $\mathbb{N}$, so on this site we say "positive integers" or "nonnegative integers" instead. A proper subset, $A \subsetneq B$, is a subset that is not all of $B$.

Why it works

Why is $\varnothing \subseteq A$ for every set $A$? The definition says: for every $x$, if $x \in \varnothing$, then $x \in A$. The hypothesis $x \in \varnothing$ is never true, so the conditional is vacuously true, exactly as in the lesson on conditionals. In the same way, every set is a subset of itself.

Worked examples

Worked example: From set-builder to roster

List the elements of $\{x \in \mathbb{Z} : x^{2} < 10\}$ and give its size.

We need integers whose square is less than $10$: $0^{2} = 0$, $1^{2} = 1$, $2^{2} = 4$, $3^{2} = 9$ all work; $4^{2} = 16$ does not.
Negative integers count too: $(-3)^{2} = 9 < 10$.
The set is $\{-3, -2, -1, 0, 1, 2, 3\}$, with $7$ elements.

Answer: $\{-3, -2, -1, 0, 1, 2, 3\}$; size $7$

Worked example: Element or subset?

Let $S = \{1, \{2, 3\}, \varnothing\}$. Decide: $2 \in S$; $\{2, 3\} \in S$; $\{2, 3\} \subseteq S$; $\varnothing \in S$; $\varnothing \subseteq S$; $|S|$.

The elements of $S$ are exactly three things: the number $1$, the set $\{2, 3\}$, and the empty set.
$2 \in S$ is false: $2$ is inside an element, not an element itself. $\{2, 3\} \in S$ is true: it is listed.
$\{2, 3\} \subseteq S$ is false: that would need $2 \in S$ and $3 \in S$.
$\varnothing \in S$ is true (it is listed), and $\varnothing \subseteq S$ is true (as for every set). $|S| = 3$.

Answer: false, true, false, true, true, $3$

Worked example: Repeated values collapse

How many elements does $\{k^{2} : k \in \{-2, -1, 0, 1, 2\}\}$ have?

The squares are $4, 1, 0, 1, 4$.
A set lists each element once: $\{0, 1, 4\}$.
So it has $3$ elements, not $5$.

Answer: $3$

Common mistake

Mixing up $\in$ and $\subseteq$, and $\varnothing$ with $\{\varnothing\}$. $1 \in \{1, 2\}$ but $\{1\} \subseteq \{1, 2\}$: $\in$ relates one element to a set, $\subseteq$ relates a set to a set. And $\varnothing$ has no elements, while $\{\varnothing\}$ has one element, the empty set (like an empty box versus a box holding an empty box). Check: for $x \in A$, look for $x$ in the list of $A$; for $X \subseteq A$, check every element of $X$ against that list.

Sets and Set-Builder Notation

Worked examples

Practice

Builds on

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Sources and further reading