In the attached youtube video, at 13:12, the lecturer gives a footnote on the difference between `Such that`

and `Subject to`

when expanding the initials mathematical abbreviation `S.T.`

in certain mathematics problems.

He claims that some people read this as `Such that`

and some as `Subject to`

where the former is a common mistake. In his words: "subjugate to followed by restriction is not the same as such that, restrictions" .

Can anyone elaborate why `such that, restrictions`

, as the lecturer states it, is wrong? In general, which is the correct form and where lies the difference between the two?

## Best Answer

tl;dr : Use either one, unless you have a specific stylistic issue with one.

Here's what professor Stephen Boyd says, adjusted slightly to make the spoken version more readable.

and

Prof. Boyd can't dismiss the usage entirely, because he knows that other members of the mathematics community use

such that, but he draws a stylistic line that urges caution ("the yellow range"), and he makes a linguistic argument roughly like the following:such thatrestriction," as an expression, does not exist in English usagesuch thatrestriction" violates rule 1 andI have two objections to this that will also answer your question. Each objection demonstrates that both usages are correct from the standpoint of language usage, even if one entails a certain risk of frustrating someone who shares Prof. Boyd's pedantry.

Such thatcan be used in English to indicate a logical restriction.Such thatacts as a conjunction. One use is to indicate the logical result or extent of an event:This usage is close to

so that.Such thatis also used in English to indicate a condition or restriction. As the answer by Hellion here states,such thatexpresses "HOW something is to be done." A mathematical restriction pertains to one meaning ofhow: the solution(s) of x (the results) must fit the conditions or restrictions; that is how it has to be solved.This usage works even outside of math. Oxford Dictionaries gives this sentence as an example for

such that:Constituting a narrativeis a condition or restriction on how sentences are linked. They explain in part how I should link sentences. I need to be aware of how the linking builds a narrative. So English does usesuch thatto set restrictions, and therefore similar usages should be valid in math.Such thatis established mathematical usage irrespective of 1.Such thatis widely used in the mathematical sciences to introduce conditions, properties, and restrictions that must be satisfied. Wolfram weighs in:This usage was standardized by the time Bertrand Russell wrote

The Principles of Mathematics(1903), a philosophical study of mathematics that required him to delve deeply into mathematical terminology. He writes a section on usingsuch that, which begins:Such that ϕx is truesets a condition or restriction on the values of x being found.I've found similar advice in writing guides for mathematics, as a search for "such that" and math will turn up. This one, provided by authors Isaiah Lankham, Bruno Nachtergaele, and Anne Schilling at UC Davis (2007), explains "such that":

Common collocations of "such that" include "there exists" and "there is," as seen in this sample:

Nonetheless, the expression

such thatdoes not have a necessary structure, and can be applied to results or conditions. So it is an accepted mathematical usage that can be used for s.t.What about? It's almost synonymous. This article never differentiates between the two:subject to?And the top answer on what

subject tomeans in this question resorts to usingsuch thatto explain its meaning:It might be possible to differentiate them more strictly through some rule where

subject toalways defines constraints andsuch thatalways gives results. I think that's what Prof. Boyd would prefer, and there may be an argument for precision in that. However, that's not how English works, and that's not how mathematics currently uses the two terms.