Editing Set-builder notation (section)

== More complex expressions on the left side of the notation ==

An extension of set-builder notation replaces the single variable {{math|''x''}} with an [[expression (mathematics)|expression]]. So instead of  <math>\{ x \mid \Phi(x)\}</math>, we may have <math>\{ f(x) \mid \Phi(x)\},</math> which should be read
:<math>\{ f(x) \mid \Phi(x)\}=\{y\mid\exists x (y=f(x)\wedge\Phi(x))\}</math>.

For example:

* <math>\{2 n \mid n \in \mathbb N\}</math>, where <math>\mathbb N</math> is the set of all natural numbers, is the set of all even natural numbers.
* <math>\{p/q  \mid p,q \in \mathbb Z, q \not = 0 \}</math>, where <math>\mathbb Z</math> is the set of all integers, is <math>\mathbb{Q},</math> the set of all rational numbers.
* <math>\{ 2 t + 1 \mid  t \in  \mathbb Z\}</math> is the set of odd integers.
* <math>\{ (t, 2 t + 1) \mid  t \in \mathbb Z\}</math>  creates a set of pairs, where each pair puts an integer into correspondence with an odd integer.

When inverse functions can be explicitly stated, the expression on the left can be eliminated through simple substitution. Consider the example set <math>\{ 2 t + 1 \mid  t \in  \mathbb Z\}</math>. Make the substitution <math>u = 2t + 1</math>, which is to say <math> t = (u-1)/2</math>, then replace ''t'' in the set builder notation to find 
:<math>\{ 2 t + 1 \mid  t \in \mathbb Z\} = \{ u \mid  (u- 1)/2 \in  \mathbb Z\}.</math>