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I am a bit confused about the semantics (or maybe it would be better to call it semiotics) of the congruence modulo. When we are presented with an expression of the form $$ a \equiv b\ (\textrm{mod}\ n)$$ this latter should be equivalent to the expression $$\exists\ c \in \mathbb{Z}\ : a - b = c\times n\ $$ Is it or is it not, and if not, why? If yes, can I use the equivalence in a formal proof?

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They are equivalent. Usually the former is defined as shorthand for the latter. As such in proofs you may cite this fact.

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  • $\begingroup$ Wonderful, cheers for the quick answer! $\endgroup$ – eslukas Feb 2 '15 at 1:59
  • $\begingroup$ Addendum: I will accept it when the 11 mins have passed. $\endgroup$ – eslukas Feb 2 '15 at 2:00
  • $\begingroup$ I meant what you meant with the existence operator. I just forgot to add it. I am editing it now. $\endgroup$ – eslukas Feb 2 '15 at 2:02
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    $\begingroup$ Ah, okay. I just thought I should be thorough. $\endgroup$ – dalastboss Feb 2 '15 at 2:06
  • $\begingroup$ Don't worry, I am very grateful that you pointed it out, tidiness is quite important in proofs. $\endgroup$ – eslukas Feb 2 '15 at 2:08

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