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How to deal with the following contradiction in notation?

$\bigcup a$ may mean both:

  • the union of a collection of sets $a$;
  • $\bigcup_{i\in \operatorname{dom}a} a_i$ for an indexed family $a$ of sets.

I deal with mathematics long time, but discovered this contradictory notation only a few minutes ago.

What is the right way to deal with this?

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    I agree with Andres. Although, as Nate says, one might write $\bigcup a_i$ without making explicit the set $x$ over which $i$ ranges (if $x$ is clear from the context), I would regard the notation $\bigcup a$ (with no subscript) for this union as simply wrong.2012-12-24

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If you have dealt with mathematics for a long time, this should not be the first time you come across the same notation used for different things. The reason is usually that there are 'too few' symbols for 'too many' mathematical concepts. Worse still, consider:

$|x - 2|y|z + w|$

Does it mean "$|( x - 2 \cdot |y| \cdot z + w )|$" or "$|x-2| \cdot y \cdot |z+w|$"?

Would you then say that there must be a right way to handle this problem with absolute value notation? Or that we should not use this notation at all?

The goal of mathematical writing is usually to convey mathematical ideas to the reader, so if that is accomplished we often do not care too much for absolute syntactic consistency.

A frequent example is when an author says halfway through:

From now on we shall drop subscripts when they are clear from the context.

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    Personally I follow Andres's recommendation of having all notation clearly and unambiguously defined, but we should still be able to read and understand writings that are not so systematically precise.2016-03-02
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The notations are not contradictory because they are syntactically different. In fact you can define one in terms of the other: $ \bigcup_{i\in I} F(i):=\bigcup\{\,F(i)\mid i\in I\,\}$


Actually, we deal with worse cases than this: Consider $\sum_{i=n}^ma_i$ which is the sum of $a_n,\ldots, a_m$ (first summand $a_n$, last summand $a_m$), whereas the syntactically similar notation $\sum_{i=n}^\infty a_i$ does not stand for a sum with a "last summand $a_\infty$" and doesn't even stand for a sum at all (it is a series instead). One may justify this by saying that the symbolism $\sum_{\square_1=\square_2}^{\square_3}\square_4$ has no meaning per se, but if $\square_2$ and $\square_3$ are replaced with elements of $\Bbb Z$ then we have a sum; this leaves us freedom in defining the symbolism independently for any cases where non-elements of $\Bbb Z$ are plugged in, for example for the case where $\square_3$ is replaced with the symbol (not number) $\infty$.