I struggled to understand where "denote" is better than "be". For example:
Let A denote a vertex cover of the graph.
or
Let A be a vertex cover of the graph.
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2If you want to sound pseudo-smart, use denote as per your example. If you want to be easily understood by everyone (and you're not a poser), use be.– FumbleFingersCommented Aug 20, 2023 at 23:39
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4@FumbleFingers - what's 'pseudo-smart' about being aware of a difference between being something and denoting it?– Michael HarveyCommented Aug 21, 2023 at 13:30
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2@MichaelHarvey: The average learner who might ask about this difference would kinda have to be "smart" to understand, remember, and correctly implement the choice about when it's "valid" to use denote. But as ruakh points out, Note that even when "denote" is appropriate, "be" is still fine, and often preferable. So arguably the really smart choice is to interpret to denote as to represent, to stand for, to be if you read it from someone else, but to stick with to be in your own writing. Less to remember, and you'll never actually be "wrong".– FumbleFingersCommented Aug 21, 2023 at 13:53
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I disagree with the answers. Whilst in technical contexts, 'denote' may be the "introduction of unique etc ...", but in general usage it means represent, or indicate. Similarly, in general usage, 'be' means is. 'A" IS NOT a vertex blah blah, "A" is a letter in the alphabet. If you don't want to tread on technical toes, avoid 'denote', but I'd use 'indicate' or 'represent' or other similar words in place of 'be'.– mcalexCommented Aug 24, 2023 at 0:06
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2 Answers
Your example should use be.
We generally use the verb "denote" only if we're introducing a notation for an existing unique entity; hence "Let <notation> denote <entity>" nearly always involves an entity introduced with the, not a. For example:
- "Let S denote the set of all sets that are not elements of themselves." (From A Book of Set Theory, page 3.)
- Of course, this example is being used to demonstrate that that's not actually a well-defined set; but that ultimate conclusion is not relevant to the choice of the verb "denote".
- "Let 1 denote the category with one object, called 1, one identity morphism id1, and no other morphisms." (From An Invitation to Applied Category Theory: Seven Sketches in Compositionality, page 94.)
So in your example, "denote" isn't appropriate, because any non-empty graph has multiple vertex covers, and we're choosing one of them arbitrarily; we can't just start "denoting" it without deciding which one it is. (If you really wanted to use the verb "denote" here, I suppose you could separate the "choose a vertex cover" step from the "define a notation" step — something like "We choose some vertex cover of the graph, and denote it by A" — but there's just no reason to do that.) By contrast, the formula "Let <symbol> be <characterization>" is a very standard way to combine selecting an entity with assigning it a name.
Note that even when "denote" is appropriate, "be" is still fine, and often preferable; for example, I'd expect to see "Let x be the greatest element of S", rather than "Let x denote the greatest element of S", even though there's nothing really wrong with the latter.
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1I'd never have thought of which article normally follows Let X denote... - but as that chart clearly shows, it's far more likely to be the definite article (for exactly the reason you give, I'm sure). Commented Aug 21, 2023 at 10:28
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2@FumbleFingers I’d expect to see denote the if we’re denoting something unique, and denote a / some if we’re choosing arbitrarily.– DavislorCommented Aug 21, 2023 at 12:24
In the context of scientific work, "denote" means that you are introducing notation. For example:
A circumflex denotes the unit vector in the direction of a vector: $ \hat{x} = \frac{\vec{x}}{\left|\vec{x}\right|} $
Notice that you absolutely cannot say "A circumflex is the unit vector...", which makes no sense. When you are giving the complete name or symbol of the thing you want to denote, you can (and probably should) use the simpler word "is". For example:
$\hat{x}$ is the unit vector in the direction of $\vec{x}$.
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1For your last example, I would still use "denotes", assuming I intended to imply that the same notation works if x is replaced by something else. But in that case I would not be giving the complete symbol, just one example, so I think that still fits your rule. Commented Aug 22, 2023 at 10:46