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?
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2$X$ and $Y$ are independent of each other. – Brian M. Scott Jan 14 '15 at 15:45
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$\lnot(x \leftrightarrow y) \equiv \lnot(x\rightarrow y) \lor \lnot (y \rightarrow x)\not\equiv \lnot(x \rightarrow y) \land \lnot(y \rightarrow x$. Note that this is a disjunction of negations, not a conjunction. So the negation/opposite of $x\leftrightarrow y$ does not forbid one from implying the other; it forbids that both $x, y$ imply the other. – amWhy Jan 14 '15 at 16:02
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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, $$X \oplus Y \equiv (X \land \lnot Y) \lor (\lnot X \land Y)$$
amWhy
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It is not clear to me that the original question is about propositional logic. I took $X$ and $Y$ to be sentences in natural language, rather than Boolean variables. – Carl Mummert Jan 15 '15 at 11:40
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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 $\exists m\in \Bbb{N} : n = 2m$
Mark Fischler
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