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I am learning some Discrete Math and was wondering whether, in a compound statement, I can use "where" and "such that" interchangeably without any problem?

For example, I am quite sure that

  • $ax^2+b=7$ such that $a,b\in\mathbb C$

and

  • $ax^2+b=7$ where $a,b\in\mathbb C$

mean the same, but is interchanging "where" and "such that" in a compound sentence always valid?

Also, is there is a symbol for "where" or "such that"? I just use ":" for such that.

ryang
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  • I'm afraid I wasn't able to understand the question you were asking, and I suspect others might feel that way too. Can you try rewriting your question? – Elchanan Solomon Aug 05 '22 at 07:36
  • I would refer to $2+2=4$ as "this equality", not "this compound statement". Also, the example you gave seems more problematic in wording than anything having to do with whether "where" or "such that" is used. For example, when you say "there is a $2$", it appears that you are singling out an unspecified choice of one of the two $2$'s on the left side of the equality (i.e. the reference is to a specific character in the ordered sequence of five characters that form "$2+2=4$"), but when you say "sum of two "2"s is", you seem to be talking about the positive integer $2.$ – Dave L. Renfro Aug 05 '22 at 07:44
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    I suppose this might be more of a linguistical question than a maths question. English is not my native language, but to me "where" and "such that" are not interchangable. My understanding of your example might enlighten someone (or just confuse everybody) but here goes: With "where" your example is a simple statement of a fact, with "such that" you imply that the sum of two 2's is 4 because there is a 2 in the equality at the start? – Henrik supports the community Aug 05 '22 at 08:24
  • Thanks so much for commenting and the first answer was spot on with what I was looking for. The wording was pretty bad on my part so that also answers questions but here is a more simple example to clarify. Suppose we have the following compound statements- "Two plus two equals four such that (if two plus x = 4 then, x =2)." and "Two plus two equal four where (if two plus x = 4, then x = 2)". –  Aug 05 '22 at 20:11
  • @ryang Are you and Saitamatugasan the same person? Why would put words in someone else's mouth and change the question so much that it is difficult to recognise? – paperskilltrees Aug 06 '22 at 04:17
  • Your replacement example was still wonky because the sentence's first part doesn't have an $x$ to link to its second part. I've edit a clear and better example into your Question; hope it suits. (It's okay to remove the broken examples since they haven't been addressed in any Answer.) – ryang Aug 06 '22 at 04:18
  • @paperskilltrees Our comments are crossing. I have not changed the spirit of the post, just fixed it for clarity (it was initially receiving downvotes presumably for being incoherent), and changed the example (and only because the OP tried to repair their initial one and still failed), which, as communicated above, the OP can revert if it doesn't suit. Cheers. – ryang Aug 06 '22 at 04:27
  • You could substitute "and" for both conjunctions. The meaning would be more clear IMHO. – Dan Christensen Aug 06 '22 at 19:59
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    I would use these words this way: (1) Choose $a,b\in\Bbb C$ such that $ax^2+b=7$. (2) The parameter $x$ satisfies the equation $ax^2+b=7$ where $a,b\in\Bbb C$. – Mateo Aug 09 '22 at 02:35

1 Answers1

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  1. The word ‘where’ (‘wherever’/‘whenever’/‘when’) sometimes means if:

    $$P(x)\text{ where }Q(x)$$ might mean $$\textbf{if }Q(x),\text{ then }P(x).$$ Example: $\text“x^2>0$ where $x\in\mathbb R.\text”$

  2. The word ‘where’/‘with’ (‘such that’/‘for which’) sometimes means and:

    $$\ldots P(x),\text{ where }Q(x)\\\ldots P(x)\text{ with }Q(x)$$ might mean $$\ldots P(x)\textbf{ and }Q(x).$$ Example: “Assume that $x$ is a perfect square, where $x$ is even.”

  3. The word ‘where’/‘with’ sometimes means for every or for some:

    $$P(x),\text{ where }Q(x)\\P(x)\text{ with }Q(x)$$ might, respectively, mean $$P(x)\textbf{ for every }Q(x)\\P(x)\textbf{ for some }Q(x).$$ Example: $\text“b=x^2$ with $x\in\mathbb Z.\text”$

So, I agree with Paul Halmos that the word ‘where’ (and the word ‘with’), when used to introduce a clause in mathematical/technical writing, should be deployed sparingly, to avoid ambiguity.

Here is an example in which a single occurrence of the word ‘where’ is confusingly doing double duty to alternately invoke both Meaning 1 (implication) and Meaning 3 (existential quantification)!

ryang
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    Thanks so much and way more clear! –  Aug 05 '22 at 20:12
  • Good answer, but I can't agree that the difference between a restrictive and a non-restrictive clauses is "tiny and technical". It is seemingly tiny, but technically huge. – paperskilltrees Aug 06 '22 at 04:35
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    @paperskilltrees Thanks. The thesis of my Answer is of course that the difference that you mention does matter. On the other hand, in the above, I am pointing at "the difference between between sentences $(1)$ and $(2)$", i.e., that little comma, not the clauses that succeed/follow it. $\quad$ Even if readers and writers do respect and observe such a comma, while parsing that sentence, an error allowance is natural and reasonable: to error-correct away its existence or to error-correct it into existence. – ryang Aug 06 '22 at 08:59
  • @paperskilltrees On reconsideration, I think that both usages of "where" introduce restrictive clauses, so have edited out that bit about restrictive-nonrestrictive clauses. – ryang Aug 09 '22 at 04:55
  • @ryang First, I believe we're dealing with restrictive clauses in (1) and (3.), and a non-restrictive clause in (2). In my understanding, the key difference between the two types is that a non-restrictive clause expresses a falsifiable statement (an assertion) at the level of conjunction: as we can imagine an hierarchy of nested clauses, where a non-restrictive clause is a part of a restrictive clause, which is inside a non-restrictive clause and so on. This hierarchy is why $\exists x (P(x) \land Q(x))$ is restrictive, while the equivalent $P(x),\text{ where }Q(x)$ is non-restrictive. [1/n] – paperskilltrees Aug 09 '22 at 15:25
  • @ryang ... Once you start a statement with a quantifier, everything becomes restrictive on the outside, but the inside is unaffected. Second, your usage of "such that" in (2) sounds wrong to my ear. Third, I don't like your decision to remove the mention of restrictive and non-restrictive clauses. In my opinion, this is the central issue, as it deals with the logical / mathematical meaning of statements. Understanding why certain conjunctions are suitable or unsuitable is a prerequisite to understanding whether one can be swapped for another. Thanks for keeping me updated. [2/2] – paperskilltrees Aug 09 '22 at 15:45
  • Thanks for the clarification! I changed my name from "saitamatugasan" to "who I am" but originally my math was pretty bad and didn't see the clear distinction between "such that" and "where" till now.... However, after I combed though 1.1 and 1.2 for 3 different books, I disagree that "IF" is equivalent to "WHERE" or "AND" because both "WHERE" and "AND" imply that something is before them. Suppose, "IF A=B, then B=T" and "And A=B, then B=T" and "Where A=B, then B=T". I agree that "WHERE" is equivalent to "AND" but "IF" has a lack of "bijection?" in comparison to "AND" and "WHERE". –  Sep 24 '22 at 14:35
  • @ryang Also, I argue (and sort of agree with you) that the comma used for "WHERE" in (2) doesn't need to be there for logic but it does make it sound better linguistically so not sure why a comma is needed where its job is to separate the two propositions. For a higher order if-then logical connective, it makes sense because we can expand that to ~P or Q where the "separation" is more clear but there is still a connection. ( P, or Q ), (P, and Q), (P, but Q), (P, where Q), (P, ~Q) are just separations without connections. –  Sep 24 '22 at 16:19
  • @ryang I am more just nitpicky on preference with the IF but with (1.), (P(x)=T and Q(x)=T)... Then that's a statement that can be either True or False ... True if x is the correct initial condition and False if x is not. I am still a bit new but would ThereExistsx (if x is True, then P(x) =T and Q(x)=T) be a bit better since x is what determines whether P or Q are T/F? –  Sep 24 '22 at 16:52
  • @ryang Wait, did you answer my "such-that vs where" question? if so then (1.) is wrong, (2.) is ok, and (3.) is good. (1.) is wrong because 'if' is not logically equivalent to 'where' xor 'and' ... assuming in "(1's)" perspective, both 'and' and 'where' are not logically equivalent; however, they are unless its a spelling issue so I agree with (2.). If you mean that "'if' sometimes means 'where' to English/programmers/unicorns" then I would assume your correct but it will never mean that for logic/math. forget about my hoobla comments this is my stance. –  Sep 28 '22 at 03:03
  • @whoIam I'm not comprehending your comment; however, do note that this discussion was never about literal translations but concerened only with the best propositional-logic capture in each case. For example, the word 'but' means "and" in propositional logic, and additionally signals contrast. – ryang Jan 29 '23 at 10:41