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?


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

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.