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I can't find this anywhere, but a particular lecturer that I have seems to be using commas to mean "such that" in sets. I'm seeking clarification on whether this really is the case or if I'm just misinterpreting. Example: $$\{U_1 \times U_2, U_1 \textit{open in }X_1, U_2 \textit{open in }X_2\}.$$ Is this the same as $$\{U_1 \times U_2| U_1 \textit{open in }X_1, U_2 \textit{open in }X_2\}?$$

Irregular User
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  • That's probably what they mean. It's unfortunate about the first comma, though if the second is meant to be interpreted as a logical "and", that's not so bad. – parsiad Feb 17 '16 at 16:30
  • The thing is, it's not a one time thing, The lecturer has been doing this over the whole course, switching between using $,$ $|$ and $:$ so I wanted to know if using $,$ is actually "correct". – Irregular User Feb 17 '16 at 16:34
  • The usage is not uncommon, in both mathematical English and ordinary English. For example, one can write $\Pr(A,B \mid C$ to mean the probability of $A$ and $B$, given $C$. – André Nicolas Feb 17 '16 at 16:43
  • @André Nicolas Just to clear that up, I don't mean the usage of $,$ to mean "and", but to mean "such that". – Irregular User Feb 17 '16 at 16:45
  • If the lecturer is speaking as he says this, and explains what he means, good... But when it is written, not so good. – GEdgar Feb 17 '16 at 16:49
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    This usage is not the most common, but it doesn't seem wrong, either. It's similar in meaning to the first comma in "Let $a_i = i^2, i = 1,\dots, n$" or "the numbers $i^2, i = 1, \dots, n$." If you're allowed to write things like ${U_1 \times U_2}$, then there's no clear reason a comma like that one shouldn't be allowed inside the braces without a $\mid$. – David Feb 17 '16 at 17:03
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    It should be noted that the meaning of $\mid$ in a notation like ${x^2 \mid x \in \mathbb{R}}$ is fundamentally different from what it means in one like ${x \mid x^2 + 5 = 9 }$. The first is short for ${y \mid \exists x \in \mathbb{R} , y = x^2}$. I wouldn't be surprised if some people objected to using $\mid$ in cases like the first, and might prefer to use some other symbol to distinguish the two concepts. Your example with ${ U_1 \times U_2 }$ is an example of the first kind. – David Feb 17 '16 at 17:13

3 Answers3

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It is not wrong to use "$,$" for "such that" in this context (however, it is not usual). In fact, from the logical point of view, we can use any symbol to denote anything (after all, truth is invariant under changes of notation). The point is: the notation's meaning have to be at least (i) clear for the reader and (ii) unambiguous.

In the first set in your post, the symbol "$,$" has two different meanings (as explained in the @par 's comment). So, I'd say that the comma's use is being inappropriate. But, of course, for the experienced reader this use will not cause problems.

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. (Alfred North Whitehead) - From Tao's blog.

Community
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Pedro
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Your interpretation is correct. The usual (and better) notation is $$\{U_1 \times U_2:\, U_1 \text{ open in }X_1 \text{ and } U_2 \text{ open in }X_2\}.$$

John Bentin
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  • Rather than knowing if my interpretation is correct, I'm interested in knowing if the lecturer is correct in using such notation. – Irregular User Feb 17 '16 at 16:35
  • @IrregularUser: In short, no. But there is an element of opinion regarding notation. What one person regards as accurate, another will consider cumbersome and time-consuming, like "Come on; you know what I mean. Don't be pedantic." – John Bentin Feb 17 '16 at 17:03
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No one can say that it's wrong to use a comma like that since we might just not have seen the convention your teacher is using, but I've never seen a convention where you can use a comma like that. Either "$:$" or "$|$" would be fine in place of the comma.

Oscar Cunningham
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