set-builder notation

1. (set theory) 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\}}
     

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  Set-builder notation on Wikipedia.Wikipedia
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  Intension on Wikipedia.Wikipedia
• "Set-Builder Notation" in Mathwords.