What does $\not\equiv$ mean?

Question

I understand that '$\equiv$' can mean "identical to" or "defined to be equal to", so intuitively, I assume that the $\not\equiv$ means "is not identical to" or "defined to be not equal to".

Am I right?

Answer 1

$\not\equiv$ 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 ($\phi \not\Leftrightarrow \psi$)

Answer 2

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 $\equiv$".