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I have some notation questions on the following equation solving:

2+x=5$\iff$x=5-2$\iff$x=3

Would you read the above as "two plus x equals five if and only if x equals five minus two if and only if x equals three" ? Somehow it doesn't sound natural to chain more than one "if and only if".

What's the notation to use and how do you read it when you are not interested in the "reverse direction". E.g. I want to write something like:

x=2 $\implies\ x^2$=4

... where using $\iff$ would be a mistake. Is the $\implies\ $ symbol acceptable and how do you read it?

Marcus Junius Brutus
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    I read the first as "is equivalent to", and the next as "which is equivalent to", if I bother to verbalize them at all. The symbol "$\Rightarrow$" is read "implies". – David Mitra Jan 14 '14 at 21:19
  • Following David's comment above: the code to make the $\implies$ arrow is \implies, which serves for verification and future reference. – Ben Grossmann Jan 14 '14 at 21:21
  • It's worth noting that in formal logic, $p \leftrightarrow q \leftrightarrow r$ does not actually mean $(p \leftrightarrow q) \text{ and } (q \leftrightarrow r),$ but rather, $p \leftrightarrow (q \leftrightarrow r).$ They aren't equivalent. (In mathematics though, $p \Leftrightarrow q \Leftrightarrow r$ is indeed safely verbalised as per David's above suggestion.) – ryang Nov 19 '21 at 17:39

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The statement $a\iff b$ ($a$ "if and only if" or "iff" $b$) is equivalent to saying $a \implies b$ ("$a$ implies $b$") and $b \implies a$ ("$b$ implies $a$"). So, $2+x = 5 \iff x = 5-2$ means that if $2+ x= 5$, $x = 5-2$ and if $x = 5-2$, $2+x=5$. This isn't usually necessary in proving something like what you have here (I usually just resort to $2+x = 5 \implies x = 5-2$), but it is to make clear that each statement implies the other.

Clarinetist
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