# Teaching the proper syntax of “such that”

I'm teaching a college course in discrete math, where students are writing their first proofs. One of the phrases we use a lot in mathematical English is such that. I notice that students don't seem to know when it should be used and sprinkle it randomly in their proofs.

Here's an example:

For a and b to be relatively prime, gcd(a,b) = 1 = ax + by s.t. x, y are integers.

My first comment is that two equivalent conditions are being asserted at the same time, which seems sloppy, but the major issue is with such that $x$ and $y$ are integers. I wrote, "'such that' is a phrase that modifies a verb of being, e.g., 'exist'. Putting it after 'equals' doesn't really sound right. You could say, '$1 = ax+by$ where $x$ and $y$ are integers,' or '$1 = ax + by$ for integers $x$ and $y$.'"

I don't think many of my students know a lot about English grammar, let alone the roughly half of the class who are ELLs. So other than modeling proper writing, what be useful ways to explain this?

• Too short for a proper answer, but I want to say "=" is already a verb of being. It's pronounced "is equal to". Still, I agree with you that "where" or "for" would be better in this context since you're just saying what you mean by x and y rather than specifying conditions on x and y. – DavidButlerUofA Dec 3 '17 at 22:25
• @DavidButlerUofA You are right about is equal to being a verb of being. So it's something else. But I can't put my finger on it. – Matthew Leingang Dec 4 '17 at 0:03
• Is it that “such that” has to go with a situation of choice? In set notation you are choosing the elements such that they satisfy a condition. When you state a theorem it’s about choosing those that apply from among all those possible. – DavidButlerUofA Dec 4 '17 at 1:00
• @ DavidButlerUofA: Wouldn't your example from set theory show that it's not a matter of choice, but condition? – SCS Dec 4 '17 at 4:14
• Maybe related: english.stackexchange.com/questions/16883/… – Matthew Leingang Dec 4 '17 at 16:25

I suggest having a class discussion about formal logical phrases and symbols, and how to translate those into natural English. For example, your student wrote this

For a and b to be relatively prime, gcd(a,b) = 1 = ax + by s.t. x, y are integers.

$\forall a,b\in \mathbb{Z}\large{\textbf{.}}\normalsize (a\text{ and }b\text{ are relatively prime}) \iff \exists x,y\in\mathbb{Z}\large{\textbf{.}}\normalsize ax+by=1$

Show them both and have the class discuss the ways in which the two versions are similar and dissimilar. The point of this discussion is not to clarify that the student's version is "wrong", nor that the other version is "more correct". Rather, the point is to have students think at all about how their mathematical writing comes across to others who read it, and to recognize that there are often many ways of saying the same thing, and some can be clearer or less ambiguous than others.

When I teach an intro to proofs course, the first month or so is a combination of content (sets, proof techniques) and an introduction to quantifiers and logical symbols. We practice translating between "wordy sentences" and "mathy/symbolic sentences", like the two examples above. In the process of translating symbolic sentences into natural English language sentences, we collectively realize what I wanted them to learn all along: "such that" comes after an existential quantification.

Another suggestion, "such that" is perfectly synonymous with the wordier but perhaps more natural phrase "with the property that". Perhaps you can exclusively use that one with your students for a while until they use it comfortably and correctly.

Addendum: In fact, I have a $\LaTeX$ macro for the "large dot", and the shorthand is \st for "such that": \newcommand{\st}{{\text{\huge {.}}}\;}

Edit: If you have not used the logical symbols $\forall$ and $\exists$ and they are not part of your course learning objectives, you can replace them with the words "for every" and "there exists". Again, the motivation behind comparing the two versions is not "formal vs. informal" necessarily, but rather striving for correctness, clarity, and a lack of ambiguity.

• "Surely, their intended message was this:" This sentence feels overly optimistic. This what you know they should have meant, but probably not what they thought. The formulation seems to indicate they where to express only one direction in Bézout's theorem, and more importantly I would assume they used "s.t." while meaning "where", following a current habit of introducing variables a posteriori. The problem is that they are usually not able to interpret this correctly (i.e. replace "where" by the appropriate quantifier and in front of the term). – Benoît Kloeckner Dec 6 '17 at 21:35
• @BenoîtKloeckner: Yes, I suppose I should have said instead, "What their message ought to have been was ..." I don't think the instructor should just write my statement on the board and say, "You meant this, right?" Rather, I am envisioning the instructor engaging a discussion where they guide the student into realizing that this is what they should have thought, and how they should have expressed that thought. Most of the learning will happen in this discussion, not as a direct consequence of showing them what would have been better. – Brendan W. Sullivan Dec 6 '17 at 21:53
• Your use of the bold point is a bit troubling. In the second highlighted sentence the first dot does not read as "such that" (it is not a restriction to the universal quantifier), while the second does. How do you write when the universal quantifier is restricted to elements satisfying a condition ? – Benoît Kloeckner Dec 7 '17 at 12:56
• Also note that the spacing of this formula can be misleading: students could interpret it with the wrong parentheses $(\forall \dots \text{prime})) \Longleftrightarrow \dots$. – Benoît Kloeckner Dec 7 '17 at 12:58

One possibility is to avoid "such that" when it does not directly translate the $|$ symbol in set builder notation, such as $\{n^2 \mid n\in \mathbb{Z}\}$. There it is more or less unambiguous. For statements of theorems, one could say, "Suppose $a$ and $b$ are integers with the property $a^2+b^2$ is an integer." Or, "Consider pairs of integers $\{(a,b)\mid a^2+b^2\in \mathbb{Z}\}$," where s.t. fits in.

Upon further reflection, I don't do this myself, but maybe I should aspire to it; it might help in other ways too.

It is admittedly confusing for ELLs. But nothing to worry too much as students would pick it up automatically with enough exposure and practice.

Meanwhile you can try showing them a few examples to differentiate between "for", "such that", "there exists" etc.

More often than not, such that will be used as a condition on a proposition. So, crudely speaking (for beginners), a mathematical statement follows the structure:

(Proposition) s.t (condition) for (domain)

"There exists" would fall under the proposition itself.

• Thanks for your answer. But proper mathematical writing is one of the learning outcomes of the course, so I don't feel right in just letting it work out eventually. I will continue to give examples of proper usage, though. Your structure is interesting—do you have more examples? For instance, we use "such that" in describing sets. Does that fit your structure? – Matthew Leingang Dec 5 '17 at 14:00
• @MatthewLeingang yes, sets is what I had in mind when I proposed this structure. IMO, pedagogy may not be apt for this learning outcome. If anything, a pedagogical approach might make it even harder for ELLs to learn it. Teaching this formally might confuse them more. In any case, not every learning outcome can have a corresponding lesson plan. This is one of the outcomes that should be acquired holistically by the end of the course. – yathish Dec 6 '17 at 8:20

I would simply say that "such that" introduce a restriction or additional information. You need to give several examples and explain the differences.

1. "Let $x$ be a real number such that $x^2>1$. Then $x>1$ or $x<1$": we introduce a variable and assume some condition on it.

2. "For all real number $x$ such that $x^2>1$, $x>1$ or $x<1$": we restrict the universal quantifier (this is only a slight variant of the previous case).

3. "There exists some real number $x$ such that $x^2>1$.": we claim an additional property (this makes sense but may look backward compared to the universal quantifier).

4. "$\{x\in \mathbb{R} \mid x^2>1 \}$": we restrict the numbers we consider by using a condition.

Added in edit after comment: the above kept implicit an important point: "such that" is used when variables are introduced, and on top of the words that introduce them. In "$1 = ax+ by$ such that $x$ and $y$ are integers", the main problem is that the variables are not introduced in the first place (or rather, "such that" is used to introduce them, while it is not its purpose, making the phrase senseless).

• This is going in the right direction. But "introducing a restriction" doesn't seem to rule out "$1 = ax+by$ such that $x$ and $y$ are integers." – Matthew Leingang Dec 8 '17 at 17:11
• @MatthewLeingang: I edited to answer your remark. – Benoît Kloeckner Dec 9 '17 at 16:03