Notation or verbiage for the opposite of 'iff'?

Question

Given the statement $X \implies Y$ and $Y \implies X$, we have the common notation $X \iff Y$. Ok so is there an opposite of this concept? Suppose I have $X$ doesn't imply $Y$, nor does $Y$ imply $X$...i.e. neither implies the other. What would you say or write for this? Whats the correct word?

Answer 1

In boolean algebra notation, one can write $X=Y$ which is equivalent to saying that $X\Leftrightarrow Y$. The opposite of that is $$ X \neq Y $$

And the wording equivalent to the opposite of iff is "unless", as in

$n\in \Bbb{N}$ is even iff $\exists m\in \Bbb{N} : n = 2m$

$n\in \Bbb{N}$ is odd unless $\exists m\in \Bbb{N} : n = 2m$

Answer 2

In logic, $X\,$ XOR $\,Y$, which we can denote $X\oplus Y,\,$ is the negation of $X \leftrightarrow Y$.

That is, $$X\oplus Y \equiv \lnot (X \leftrightarrow Y)$$

There are a number of common ways in which $X\oplus Y$ is defined: $$X\oplus Y \equiv (X \lor Y)\land \lnot (X\land Y)$$

Alternatively, $$X \oplus Y \equiv (X \land \lnot Y) \lor (\lnot X \land Y)$$