Editing Equivalence relation (section)

=== Equivalence relations ===

The following relations are all equivalence relations:
* "Is equal to" on the set of numbers. For example, <math>\tfrac{1}{2}</math> is equal to <math>\tfrac{4}{8}.</math><ref name=":0" />
* "Is [[Similarity (geometry)|similar]] to" on the set of all [[Triangle (geometry)|triangle]]s.
* "Is [[Congruence (geometry)|congruent]] to" on the set of all [[Triangle (geometry)|triangle]]s.
* Given a [[Function (mathematics)|function]] <math>f:X \to Y</math>, "has the same [[Image (mathematics)|image]] under <math>f</math> as" on the elements of <math>f</math>'s [[domain of a function|domain]] <math>X</math>. For example, <math>0</math> and <math>\pi</math> have the same image under <math>\sin</math>, viz. <math>0</math>. In particular:
** "Has the same absolute value as" on the set of real numbers
** "Has the same cosine as" on the set of all angles.
** Given a natural number <math>n</math>, "is congruent to, [[Modular arithmetic|modulo]] <math>n</math>" on the [[integers]].<ref name=":0" />
** "Have the same length and direction" ([[equipollence (geometry)|equipollence]]) on the set of [[directed line segment]]s.<ref>[[Lena L. Severance]] (1930) [https://babel.hathitrust.org/cgi/pt?id=mdp.39015069379678;view=1up;seq=15 The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis], link from HathiTrust</ref>
** "Has the same birthday as" on the set of all people.