Editing Equivalence relation (section)

=== Projection ===
{{main|Projection (relational algebra)}}

The ''projection'' of <math>\,\sim\,</math> is the function <math>\pi : X \to X/\mathord{\sim}</math> defined by <math>\pi(x) = [x]</math> which maps elements of <math>X</math> into their respective equivalence classes by <math>\,\sim.</math>
: '''Theorem''' on [[Projection (set theory)|projection]]s:<ref>[[Garrett Birkhoff]] and [[Saunders Mac Lane]], 1999 (1967). ''Algebra'', 3rd ed. p.&nbsp;35, Th. 19. Chelsea.</ref>  Let the function <math>f : X \to B</math> be such that if <math>a \sim b</math> then <math>f(a) = f(b).</math> Then there is a unique function <math>g : X / \sim \to B</math> such that <math>f = g \pi.</math> If <math>f</math> is a [[surjection]] and <math>a \sim b \text{ if and only if } f(a) = f(b),</math> then <math>g</math> is a [[bijection]].