**Exists** and **exist** follow the ordinary convention for verbs: one is singular and the other is plural. Where mathematical usage differs from ordinary usage is in the way singular and plural are indicated in the *subject* that follows, and an implied “for all” later if the subject is plural.

**Exists** is singular:

There exists *s''* in *S* such that *s''*(*s'm* – *sm'*) = 0.

Or, spelled out more explicitly:

There exists a number *s''* in the set *S*, such that *s''*(*s'm* – *sm'*) = 0.

**Exist** is plural:

There exist *α*_{i} in *I* such that *x*_{n} = Σ *α*_{i} x_{i}.

This is where the mathematical usage differs from ordinary usage. Spelled out explicitly, this would be:

There exist numbers *α*_{1}, *α*_{2}, *α*_{3}, etc., which are elements of the set *I*, such that **for each** subscript, if we refer to the subscript as *i*, then *x*_{n} = Σ *α*_{i} x_{i}.

The convention is tricky for a beginner to understand because it depends on your knowing that *i* is commonly used as a variable that will stand for multiple subscripts. The plural form of “exist” is actually a helpful clue. However, you must understand the implied “for each” later in the sentence. It’s implied by the fact that the sentence uses *i* to stand for multiple values.

Related: http://ell.stackexchange.com/questions/46831/be-used-vs-to-be-used-to

– Stephie – 2015-01-16T11:57:52.753