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For example, let M and N be two real numbers. M is smaller than N. Now I negate the inequality, such that now M is greater or equal to N.

$M < N ≟ M \geq N$

Is there a sign to replace '≟' in the expression above?

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$\neg$ ${}{}{}{}$ –  Cocopuffs Aug 21 '12 at 13:27
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2 Answers

up vote 3 down vote accepted

It looks like you're talking about equivalence of a proposition to the negation of another. In which case, the "exclusive or" should do:

$$ (M < N) \;\veebar\; (M \geqslant N) $$

asserts that exactly one of the two propositions $M < N$ and $M \geqslant N$ is true. As suggested in the comments (and in Ilya's answer), though, explicitly writing one of

$$\begin{gather*} (M < N) \;\equiv\; \neg (M \geqslant N) \\ (M < N) \;\iff\; \neg (M \geqslant N) \end{gather*}$$

would probably be clearer.

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Thanks for the answer. I also liked to see the '≡' in usage. I never used it but I have been curious about it. –  GuiRitter Aug 21 '12 at 19:09
    
@GuiRitter: It is essentially standard notation for logical equivalence ($A \equiv \neg \neg A$) and congruence in modular arithmetic ($5 \equiv 12 \bmod{7}$); it is nonstandard but common for identities, ie. an implicit universal quantifier ($\cos^2(x) +\sin^2(x) \equiv 1$ or $f \equiv 0$ to describe a function which is zero everywhere), definitions of variables (in place eg. of $:=$), and perhaps similar emphatic notions of equality or similarity. –  Niel de Beaudrap Aug 21 '12 at 22:55
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$\quad\quad\Leftrightarrow \neg$

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