# What are the roles of "such" and "that" in the following sentence?

"For every number a not equal to 0, there exists a number b such that ab = 1."

I am aware of a related post on the use of "such that", I am not sure how the explanations there fit in to the use of "such that" in the above sentence.

• In mathematics, “such that ...” means “for which the following property holds: ...” Commented Sep 24, 2021 at 8:23

## 3 Answers

This is consistent with the following definition of such from Lexico

1. such — as/that Of the type about to be mentioned.
the wound was such that I had to have stitches

The "type about to be mentioned" is numbers that fit the criteria "ab = 1"

• I don't see why Lexico explains the use of "such" in my sentence. Lexico lists "such" as a determiner, a predeterminer, or a pronoun. If "such" is a determiner or a predeterminer, then what is the corresponding noun/pronoun in the sentence? If "such" is a pronoun, what is "such" referring to? Every example in Lexico involving "such that" is of the form "... is/was such that...", a pattern that my sentence does not fit into.
– aamu
Commented Sep 27, 2021 at 20:53
• The noun is a condition that `b` fits. I added one of the example sentences that is kind of similar outside the mathematical context. Commented Sep 27, 2021 at 20:57
• This isn't an explanation; it's a translation. And it's correct. What would count as an "explanation"? A bunch of logical formulae? A history of the so/such ... that `S` construction? Commented Jun 26, 2022 at 16:23

"For every number a not equal to 0, there exists a number b such that ab = 1."

I don't know what a contemporary grammarian would do with that sentence. My approach would be as follows:

1. "For every number a not equal to 0" is a prepositional phrase that presents a condition.

2. "there exists a number b", a clause that presents a fact related to the condition.

3. such... is a predicative complement of "there exists a number b", a qualifier of the fact presented, and the meaning of such is "(said number b) having the characteristic"

4. "that ab = 1" a declarative content clause which is a complement of such, stating the characteristic.

According to CGEL p. 968, "such" is an adjective licensing a that-clause complement.

• a number [such that the product is 1]: postpositive adjective phrase
• the number is [such that the product is 1]: predicative adjective phrase

As noted above, the formula "VARIABLE such that PROPERTY" is quite common in mathematical writing.