I understand that the triple equal sign can mean "identical to" or "defined to be equal to", so intuitively, I assume that the triple equal sign with a strikethrough means "is not identical to" or "defined to be not equal to".

Am I right?

  • $\begingroup$ What is the context? (I mean, what subject are you trying to learn?) $\endgroup$ – nomen Jan 15 '14 at 22:15

Yes, your intuition is correct. $a \not\equiv b $ means that "$a$ is not identical to $b$" or "$a$ is not, by definition, equal to $b$."

For a more nuanced answer, context is important. For example, $p \equiv q$ may used to express the fact that $p, q$ are logically equivalent. In another context, an expression like $a \not\equiv b \pmod n$ is used to express that $a$ is not equivalent $b \pmod n$.

For any given context in which $\,\equiv\,$ is understood to be defined, $\not\equiv$ means precisely "the negation of $\equiv$".

  • $\begingroup$ I have defined y to be a weighted sumscore of some xs. So y ≡ f(xi) I think. This makes the xs the ‘stuff’ of y. Assuming independence of xs, the xs cannot also cause y in a traditional cause-effect type way, since that would require y to be different conceptual stuff from the xs. If the xs DO cause y in a traditional cause-effect type way, then the xs cannot also be the "stuff of" y (x can’t cause itself). I want to write a general statement along the lines of “the xs are not the stuff of y”, and thought to use y ≢ f(x). Any thoughts? $\endgroup$ – John Jan 15 '14 at 23:24
  • $\begingroup$ That will indeed work! $\endgroup$ – Namaste Jan 15 '14 at 23:30

$\not\equiv$ usually denotes the opposite of $\equiv$, that is depending on the context for example

  • $f\not\equiv 0$ where $f$ is a function means that $f$ is not identical to the zero function, i.e. $$\exists x\in \mathrm{dom}(f) : f(x) \neq 0$$
  • $a\not\equiv b \bmod{n}$ where $a,b\in\mathbb Z$ and $1<m\in\mathbb N$ means that $a$ and $b$ are not congruent modulus $m$, i.e. $$m\not|\ \ b-a$$ ($m$ does not divide the difference of $a$ and $b$).
  • $\phi \not\equiv \psi$ with logical statements means that they are not equivalent ($\phi \not\Leftrightarrow \psi$)
  • $\begingroup$ BTW Thanks AlexR - have posted a 'clarification' as a comment to AmWhy's response... $\endgroup$ – John Jan 15 '14 at 23:27
  • $\begingroup$ @John in your context, $y\not\equiv f(x)$ would express that $y$ is independent of $x$; but I haven't seen this notation very often; usually it was just the statement $y$ does not depend on $x$ or something along the lines. $\endgroup$ – AlexR Jan 16 '14 at 21:10

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.