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I'm reading about congruences in number theory and my textbook states the following:

The congruence relation on $\mathbb{Z}$ enjoys many (but not all!) of the properties satisfied by the usual relation of equality on $\mathbb{Z}$.

The text then does not go into detail as to what properties they are describing. So what are the properties they are talking about? I've already showed that congruences are reflexive, symmetric, and transitive, so why in general is this not the same as equality? Is there some property that all equivalence relations will never share with the equality relation? I appreciate all responses.

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An equivalence relation is the equality relation if and only if its congruence classes are all singletons. Most equivalence relations do not have this characteristic. The equivalence classes of (most) congruence relations on $\Bbb Z$, for example, are infinite.

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  • $\begingroup$ Ahh, this does make sense, is there any other properties that equivalence relations don't share with equality or is this the main one? $\endgroup$ – Amateur Math Guy Feb 23 '13 at 15:12
  • $\begingroup$ This is precisely the property that equality doesn't share with any other equivalence relations (hence the "if and only if"). $\endgroup$ – Cameron Buie Feb 23 '13 at 17:14
  • $\begingroup$ As a belated addendum: another property that equality relations have is that they are antisymmetric. Symmetric relations (like equivalence relations) are antisymmetric if and only if they are equality relations. $\endgroup$ – Cameron Buie May 17 '15 at 14:04
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Equality is also an equivalence relation. However, equality is the finest of all equivalence relations. That is: If $\sim$ is an equvalence relation, we always have $x=y\Rightarrow x\sim y$, but the converse may not hold. In fact, for any formula $\Phi(x)$ we have that $x=y$ implies $\Phi(x)\leftrightarrow \Phi(y)$. For every other equivalence relation there must be at least one $\Phi$ violating this.

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    $\begingroup$ +1 Hagen: I hadn't ever heard it put that way: "..., equality is the finest of all equivalence relations." (I'm referring to the descriptor "finest".) Perhaps a reflection of my ignorance, but the descriptor is apropos: it pretty much says it all! $\endgroup$ – Namaste Feb 23 '13 at 16:12

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