Equivalence Relations

Definition. [Equivalence Relations] An equivalence relation is a relation that is reflexive, symmetric, and transitive.

Two objects $x$ and $y$ that are related through an equivalence relation are called equivalent. Other synonyms possible are equal, similar, congruent, and isomorphic (recall that $\text{Equal}$ and $\text{Congruent}$ are both equivalence relations!)

Possible symbols for representing equivalence relations that you may see in discrete math books are:

• $=$ ('equal'), as in $x = y$ (

  $x = y$

  )
• $\equiv$ ('congruent' or 'equivalent'), as in $x \equiv y$ (

  $x \equiv y$

  )
• $\approx$ ('approximate'), as in $x \approx y$ (

  $x \approx y$

  )
• $\sim$ ('similar'), as in $x \sim y$ (

  $x \sim y$

  )
• $\simeq$ ('sim-equal'), as in $x \simeq y$ (

  $x \simeq y$

  )
• $\cong$ ('congruent'), as in $x \cong y$ (

  $x \cong y$

  )
• $\leftrightarrow$ ('equivalent', especially from a logic perspective), as in $x \leftrightarrow y$ (

  $x \leftrightarrow y$

  )