Editing Set-builder notation (section)

== Equivalent predicates yield equal sets ==

Two sets are equal if and only if they have the same elements. Sets defined by set builder notation are equal if and only if their set builder rules, including the domain specifiers, are equivalent. That is 
:<math> \{ x \in A \mid P(x) \}=\{ x \in B \mid Q(x) \} </math> 
if and only if 
:<math> (\forall t)[ (t \in A \land P(t)) \Leftrightarrow (t \in B \land Q(t))]</math>.
Therefore, in order to prove the equality of two sets defined by set builder notation, it suffices to prove the equivalence of their predicates, including the domain qualifiers.

For example,
:<math> \{ x \in \mathbb{R}\mid  x^2 = 1 \} = \{ x \in \mathbb{Q} \mid |x| = 1 \} </math>
because the two rule predicates are logically equivalent:
:<math> (x \in \mathbb{R} \land x^2 = 1) \Leftrightarrow (x \in \mathbb{Q} \land |x| = 1).</math>
This equivalence holds because, for any real number ''x'', we have <math>x^2 = 1</math> if and only if ''x'' is a rational number with <math>|x|=1</math>. In particular, both sets are equal to the set <math>\{-1,1\}</math>.