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In definitions and exercises, I notice that "so that" and "such that" are seemingly used interchangeably. Are they in fact interchangeable, or is one more appropriate for a specific context?

Note: $\mathrm{Dom}\,(f)$ means the domain of $f$.

Example 1:

Suppose that a function $f$ is continuous at a point $c$ and $f(c) > 0$. Prove that there is a $\delta > 0$ $\color{red}{\text{so that}}$ for all $x \in \mathrm{Dom}\,(f)$, $$ |x-c| \le \delta \ \Rightarrow \ f(x) \ge \frac{f(c)}{2} $$

Example 2:

A function $f(x)$ is continuous at a point $c \in \mathrm{Dom}\,(f)$ if and only if for each $\varepsilon > 0$ there is a $\delta > 0$ $\color{red}{\text{such that}}$ for all $x \in \mathrm{Dom}\,(f)$: $$ |x - c| \le \delta \ \Rightarrow \ |f(x) - f(c)| \le \varepsilon $$

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    $\begingroup$ To me in this case “such that” seems more appropriate. “So that” to me is to be used when something is to be constructed or demonstrated to have a property, while “such that” means that we assume a property of an object. But I'm not a native English speaker, so I might be wrong. :) $\endgroup$
    – tomasz
    Aug 16, 2012 at 19:05
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    $\begingroup$ jmilne.org/math/words.html $\endgroup$
    – Andrew
    Aug 16, 2012 at 19:05
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    $\begingroup$ @David: I’ve answered some questions there; based on my experience, it would probably be suggested that this question would be better asked here. $\endgroup$ Aug 16, 2012 at 19:35
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    $\begingroup$ @Andrew: the link is indeed nice, but its author unfortunately blunders by arguing that "$a(n)\neq0$ for all $n$" should mean "some $a(n)$ is nonzero" instead of "all $a(n)$ are nonzero", which is ridiculous. $\endgroup$ Aug 16, 2012 at 19:56
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    $\begingroup$ Dear @Andrew: In my view there is no ambiguity: formulas always bind more tightly than text, so the "for all $n$" can only apply to all of "$a(n)\neq0$". Assuming that, as Milne suggests, one could take the negation bar out of the formula and make it say "NOT ($a(n)=0$ for all $n$)" is what I consider ridiculous. Note that one can't even do this is you pronounce "$\neq$" as "is unequal to". $\endgroup$ Aug 16, 2012 at 20:18

2 Answers 2

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Clauses that begin "such that" function as adjectives; they modify nouns. Clauses that begin "so that" function as adverbs; they modify verbs (or adjectives or other adverbs).

In both of the examples in the question, "such that" is the correct choice, because the clause modifies $\delta$.

In certain cases, either choice can be justified because the difference in meaning ends up not mattering. For example, in "Choose $n$ such that $2n+1$ is prime," the "such that" clause is about $n$. In "Choose $n$ so that $2n+1$ is prime," the "so that" clause is about the action of choosing; it says to do the choosing in a manner that ensures that $2n+1$ is prime. The overall effect of the two options is the same. (Personally, I'd prefer "such that" in this example.)

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  • $\begingroup$ 'So that', 'unless', 'only if' and the like all suffer from colloquial usages that frequently aren't strictly logical/correct. However, once I slow down, to my ear, 'so that' (which is synonymous with 'in order that') introduces a clause that furnishes nonrestrictive, optional information (about consequence); as such, I think that "choose $n$ so that $n+1$ is even" is technically incoherent, even as I have caught myself writing 'so that' when I actually mean 'such that'. $\endgroup$
    – ryang
    Aug 9, 2022 at 18:21
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so that

  • introduces a clause that (inessentially) conveys the intended consequence of the preceding instruction/assertion
  • logical sense: therefore
  • means ‘in order that’
  1. Let $f(x)=x^3+3x^2−10x$ so that $f(1)=−6$ and $f(2)=0.$

such that

  • introduces a (modifying) clause that stipulates a restriction
  • logical sense: and
  • variously means ‘in which’, ‘satisfying’, ‘for which’
  1. $S$ is the set of reals such that each, for some natural $k,$ equals $2k.$
  2. Every complex $z$ is such that $|z|$ is nonnegative.
  3. For each real $x$ such that $f(x)=7,$ we have that $2<x<9.$
  4. Suppose that $n$ is an integer such that $5|(n + 2).$
  5. There exists a real $c$ such that $f(c)=0.$

In the OP's given examples, ‘such that’ is correct whereas ‘so that’ is wrong.

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  • $\begingroup$ english.stackexchange.com/a/16887/172946 $\endgroup$
    – C.F.G
    Aug 9, 2022 at 16:10
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    $\begingroup$ @C.F.G Thanks; many of the Answers on that page are (seductive but) inaccurate to outright wrong. $\quad$ For example, one Answer there distinguishes the two phrases by claiming that 'such that' expresses consequence, but in fact this property is a defining feature of 'so that'! Another Answer there says that 'such that' is a description of "how", but this isn't exactly true if you consider my four examples. $\endgroup$
    – ryang
    Aug 9, 2022 at 16:27

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