"So That" vs. "Such That"

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 $x \in \mathrm{Dom}\,(f)$: $$|x - c| \le \delta \ \Rightarrow \ |f(x) - f(c)| \le \varepsilon$$

• 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. :) Aug 16, 2012 at 19:05
• jmilne.org/math/words.html Aug 16, 2012 at 19:05
• @David: I’ve answered some questions there; based on my experience, it would probably be suggested that this question would be better asked here. Aug 16, 2012 at 19:35
• @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. Aug 16, 2012 at 19:56
• 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". Aug 16, 2012 at 20:18

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.)

• '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'. Aug 9, 2022 at 18:21

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 $$2
4. Suppose that $$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.

• english.stackexchange.com/a/16887/172946 Aug 9, 2022 at 16:10
• @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. Aug 9, 2022 at 16:27