# set-builder notation

## English

### Noun

1. A mathematical notation for describing a set by specifying the properties that its members must satisfy.
• 2000, Kenneth E. Hummel, Introductory Concepts for Abstract Mathematics[1], CRC Press (Chapman & Hall/CRC), page 123:
With this idea for describing a finite set of sets, it is easy to generalize the concept to a certain infinite family ${\displaystyle {\mathcal {S}}_{2}}$ of sets ${\displaystyle {\mathcal {S}}_{2}=\{A_{i}\vert i\in N\}=\{A_{1},A_{2},A_{3},\dots ,A_{n},\dots \}}$. Once again, the power of set builder notation triumphs. The sets ${\displaystyle {\mathcal {S}}_{1}}$ and ${\displaystyle {\mathcal {S}}_{2}}$ may be described more precisely with set builder notation than by enumeration.
• 2011, Tom Bassarear, Mathematics for Elementary School Teachers, Cengage Learning, 5th Edition, page 56,
In this case, and in many other cases, we describe the set using set-builder notation:
${\displaystyle Q=\left\{{\frac {a}{b}}\vert \ a\in I\ \mathrm {and} \ b\in I,\ b\neq 0\right\}}$
This statement is read in English as "Q is the set of all numbers of the form ${\displaystyle {\frac {a}{b}}}$ such that a and b are both integers, but b is not equal to zero."
• 2012, Richard N. Aufmann, Joanne Lockwood, Intermediate Algebra, Cengage Learning, 8th Edition, page 6,
A second method of representing a set is set-builder notation. Set-builder notation can be used to describe almost any set, but it is especially useful when writing infinite sets. In set-builder notation, the set of integers > −3 is written
${\displaystyle \left\{x\vert x>-3,\ x\in \mathrm {integers} \right\}}$