# Why “facility i” but “the edge j”

I was reading an academic paper and I encountered the following sentence:

"By the assumption, facility i is fully paid before time t. For any city k, the edge j must be tight before time t."

As I know, we don't use "the" before phrases like section 2, room 5. But why in the above sentence the authors used "the edge j"? Is it correct? As a related question, in a formal writing, if I say "Let S be a set of integers". Now which of the following sentences are correct:

• The set S contains ...
• Set S contains ...
• S contains...

Let S be a set of integers. Now which of the following sentences are correct:

The set S contains ...
Set S contains ...
S contains...

None of those are grammatically incorrect, but I prefer the last one. You have already defined S as a set, so the word set is superfluous. That said, mentioning that S is a set could be warrented if it would serve as a useful reminder. For example, suppose you have a lot of one-letter variables you are discussing in one section of your paper.

Let S be the set of odd prime numbers. Let M be an 8x8 matrix such that each value is an even integer. Let H be a closed Hamiltonian graph. Let F = 96,485 J (Faraday's constant). The set S contains...

In that case, a reminder that S is a set may be justified, because single upper-case letters are being used to represent sets, graphs, constants, and matrices. A small reiteration helps the reader remember what we are talking about. However, in a paper where each capital letter denotes a set:

Let S be the set of odd prime numbers. Let T be the set of squared integers. Let R be the set of integers found in the Fibonacci sequence. S contains...

I don't see any need to specify that S is a set at the start of the sentence. However, that's a stylistic choice, not a grammatical requirement.

By the assumption, facility i is fully paid before time t. For any city k, the edge j must be tight before time t.

Here's my advice: Don't overthink it. I don't think either of these would be more grammatically correct or incorrect:

By the assumption, the facility i is fully paid before time t. For any city k, the edge j must be tight before time t.

By the assumption, facility i is fully paid before time t. For any city k, edge j must be tight before time t.

Sometimes, the inclusion of an article is absolutely necessary:

I bought car yesterday. I bought a car yesterday.

Other times, the inclusion of an article is purely optional, and mathematics provides plenty of examples of that:

Let S be the set of all prime numbers; let R be the set of all even integers greater than zero. Let T be the intersection of R and S.

The set T has exactly one member.
T has exactly one member.
Set T has exactly one member.

All three of those conclusions are stated in a way that is grammatically correct. The definite article (the) is acceptable, because there is only one set T. However, the definite article isn't necessary, because the set already has a unique name (T).

I've already explained which one of the three I prefer.

• As always, your treatment of the subject is exhaustive! – Lucian Sava Feb 18 '14 at 12:12

It sounds like each city has its own edge, and it is therefore necessary to ensure that the j under discussion is specific to the city. This requires the definite article 'the'.

If there is only one edge j, which applies to any city k, then the 'the' would not be necessary.

it is wrong, your explanation holds true.

And hence,the following are correct

Set S contains ... S contains...

However, when you have already introduced S as a set. But if you have multiple sets defined then you should be using

The set S contains ...