set-builder notation

set-builder notation

English

Noun

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, 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:

     ${\\displaystyle Q=\\left\\{{\\frac {a}{b}}\\vert \\ a\\in I\\ \\mathrm {and} \\ b\\in I,\\ b\eq 0\\right\\}}$

     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

     ${\\displaystyle \\left\\{x\\vert x>-3,\\ x\\in \\mathrm {integers} \\right\\}}$

Further reading

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