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39 lines
1.8 KiB
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39 lines
1.8 KiB
Markdown
---
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lastmod: "2021-09-12T22:47:44.0000000+01:00"
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author: patrick
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categories:
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- stack-exchange
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comments: true
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date: "2017-11-05T00:00:00Z"
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title: Abuse of notation in function application
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summary: Answering the question, "Are these examples of abuses of notation?".
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---
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*This is my answer to the same [question posed on the Mathematics Stack Exchange](https://math.stackexchange.com/q/2505777/259262). It is therefore licenced under [CC-BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).*
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# Question
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I have often seen notation like this:
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> Let \\(f : \mathbb{R}^2 \to \mathbb{R}\\) be defined by \\(f(x, y) = x^2 + 83xy + y^7\\).
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How does this make any sense?
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If the domain is \\(\mathbb{R}^2\\) then \\(f\\) should be mapping individual tuples.
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Also when speaking of algebraic structures why do people constantly interchange the carrier set with the algebraic structure itself?
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For example you might see someone write this:
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> Given any field \\(\mathbb{F}\\) take those elements in our field \\(a \in \mathbb{F}\\) that satisfy the equation \\(a^8 = a\\).
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How does this make any sense?
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If \\(\mathbb{F}\\) is a field then it is a tuple equipped with two binary operations and corresponding identity elements all of which satisfy a variety of axioms.
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# Answer
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The example you've given of a function is not an abuse. \\(x\\) is instead shorthand for \\(\pi_1(t)\\) and \\(y\\) is shorthand for \\(\pi_2(t)\\) and \\((x, y)\\) is shorthand for \\(t\\).
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\\(g \in G\\) is a very minor abuse, yes.
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"A group \\(G\\) is a set \\(G\\) endowed with some operations" is a slight abuse, but one which will never be misinterpreted.
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It is done this way to avoid the proliferation of unnecessary and confusing symbols.
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For the same reason, we use the symbol \\(+\\) to refer to the three different operations of addition of integers, rationals, and reals.
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