Import Hugo

hugo/content/posts/2017-11-05-abuse-of-notation.md

@@ -0,0 +1,38 @@
+---
+lastmod: "2021-09-12T22:47:44.0000000+01:00"
+author: patrick
+categories:
+- stack-exchange
+comments: true
+date: "2017-11-05T00:00:00Z"
+title: Abuse of notation in function application
+summary: Answering the question, "Are these examples of abuses of notation?".
+---
+
+*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/).*
+
+# Question
+
+I have often seen notation like this:
+
+> Let \\(f : \mathbb{R}^2 \to \mathbb{R}\\) be defined by \\(f(x, y) = x^2 + 83xy + y^7\\).
+
+How does this make any sense?
+If the domain is \\(\mathbb{R}^2\\) then \\(f\\) should be mapping individual tuples.
+
+Also when speaking of algebraic structures why do people constantly interchange the carrier set with the algebraic structure itself?
+For example you might see someone write this:
+
+> Given any field \\(\mathbb{F}\\) take those elements in our field \\(a \in \mathbb{F}\\) that satisfy the equation \\(a^8 = a\\).
+
+How does this make any sense?
+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.
+
+# Answer
+
+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\\).
+
+\\(g \in G\\) is a very minor abuse, yes.
+"A group \\(G\\) is a set \\(G\\) endowed with some operations" is a slight abuse, but one which will never be misinterpreted.
+It is done this way to avoid the proliferation of unnecessary and confusing symbols.
+For the same reason, we use the symbol \\(+\\) to refer to the three different operations of addition of integers, rationals, and reals.