# What does the 'triple line equals' sign with a strikethrough mean?

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?

• What is the context? (I mean, what subject are you trying to learn?) – 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 $\equiv$".

• 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? – John Jan 15 '14 at 23:24
• That will indeed work! – 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 ($\phi \not\Leftrightarrow \psi$)
• BTW Thanks AlexR - have posted a 'clarification' as a comment to AmWhy's response... – John Jan 15 '14 at 23:27
• @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. – AlexR Jan 16 '14 at 21:10