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Suppose I want to say that something is true "if and only if" <insert a series of equivalences>. Is this correct, or is it better to say "if and only if the following equation holds:" instead? This is something that has been bugging me for a while when writing Mathematics.

Version 1: The sum above is equal to $\tan(x)$ if and only if $$\frac1{u-1}=1\\\iff u=\frac{1\pm\sqrt5}2.$$

Version 2: The sum above is equal to $\tan(x)$ if and only if this equation holds: $$\frac1{u-1}=1\\\iff u=\frac{1\pm\sqrt5}2.$$

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    $\begingroup$ It doesn't really matter, especially on homework. If you were writing something up more formally you might prefer to state a result of the form "the following are equivalent: 1. Some verbal statement. 2. Some symbolic predicate. 3. etc." $\endgroup$ Commented Jun 24 at 22:26
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    $\begingroup$ Please use MathJax to typeset mathematical formula in your MSE posts. It is easy to learn and makes your posts much easier to read. Asking readers to flick back and forward between two images as you are doing here is an ask too much. $\endgroup$
    – Rob Arthan
    Commented Jun 24 at 22:26
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    $\begingroup$ Welcome to MSE. The use of pictures is highly discouraged in this forum. Please, type your questions using MathJax. Consider that many users won't even read your question if it's not properly formatted, let alone help you. $\endgroup$
    – jjagmath
    Commented Jun 24 at 22:28
  • $\begingroup$ Personally I think everyone should care more about their writing, including homework. Good on you for caring. My opinion is to write "... If and only if <first equation, inline>, or, equivalently, <the rest of the equivalences, in display form>. $\endgroup$
    – Leo
    Commented Jun 25 at 5:24

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Version 1: The sum above is equal to $\tan(x)$ if and only if $$\frac1{u-1}=1\\\iff u=\frac{1\pm\sqrt5}2.$$

Version 1 technically isn't expressing your intended meaning: since the displayed equivalence is universally true, Version 1 basically says that the above sum simply equals $\tan(x),$ without qualification.

Version 2: The sum above is equal to $\tan(x)$ if and only if this equation holds: $$\frac1{u-1}=1\\\iff u=\frac{1\pm\sqrt5}2.$$

Version 2 is clearer. Better:

  • The sum above is equal to $\tan(x)$ if and only if $\dfrac1{u-1}=1,$ that is, $u=\dfrac{1\pm\sqrt5}2.$

  • The sum above is equal to $\tan(x)$ if and only if $\dfrac1{u-1}=1$ or, equivalently, $u=\dfrac{1\pm\sqrt5}2.$

  • Now, $\dfrac1{u-1}=1\iff\ldots\iff u=\dfrac{1\pm\sqrt5}2.$

    Therefore, the sum above is equal to $\tan(x)$ if and only if $u=\dfrac{1\pm\sqrt5}2.$

  • etc.

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