# What is the proper way to use articles in formula annotations?

English is not my native language. Recently I ran into a problem while writing a scientific paper. It seems like different authors use a/the articles in formula annotations differently, and I can't understand what is the proper way to do this in academic writing. Here are two naive examples:

Example 1

(1)__ probability of operability of (2)__ unrecoverable system is defined as:
P(t) = e-λt,
where λ denotes (3)__ hazard rate.

Example 2

Let us define (4)__ mapping:
F: X -> Y,
where X denotes (5)__ set of integers, Y denotes (6)__ set of Naruto characters.
(7)__ function F maps (8)__ integer xX to (9)__ Naruto character yY.

As for me, I would fill in the blanks like this (with comments):

(1) The (specific probability, clear from the context)
(2) an (some system, not a specific one)
(3) the (it's clear from the context that we are referring to the rate λ from the formula)
(4) a (one of the possible mappings we could define)
(5) the (same as (3))
(6) the (same as (3))
(7) the (same as (3))
(8) the (the identity is clear from the context)
(9) the (same as (8))

I'm not sure about (4), (7), (8).

How to apply the rules for using articles in these cases?
I'm not sure I did it right. Please correct me if I'm wrong.

• "Please correct me" are rarely good questions. Apart from "I'm not sure" is there any particular reason you think these are wrong. Commented Mar 23 at 14:05
• @JamesK, I agree that mathematics is quite abstract and in the most situations both meaning are correct. But I believe that only one of them is the most accurate. For example, in (8), I can emphasize that I am referring to a specific subset of integers that has already been mentioned. In this case I should use "the". But maybe it's better to use "an" to emphasize that I mean some integer, one of the possible integers from X. Which meaning is more accurate in this context from the native speaker's point of view? Or maybe both are acceptable? That's what I wanted to know... Commented Mar 23 at 14:41
• That is the point I make in my answer. Does "...set of integers" mean Z, or some set X \subset Z. The choice of word depends on the meaning, not on grammatical rule. Commented Mar 23 at 15:31