# Function defined on/over/from A to B

how should one read

in plain English?

In the following sentence, for example: "let f be a function defined on/over/from/(other) A to/(other) B"?

edit:

This post has been tagged as a possible duplicate to this question. However, I originally I posted this as a followup question on the page in question, and has subsequently been deleted by a moderator:

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and someone commented:

``````Hi @Mogu ... even for related questions, you can (and should) post it as a new question.
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The question in the other post regarded the prepositions to be used with respect to the domain of a function, whereas mine is also concerned with its codomain, and whether or not the act of mentioning the codomain in a sentence changes what prepositions should be used to relate to its domain as well.

• Possible duplicate of "Function defined on/over the set A" Jun 11, 2019 at 13:16
• That symbolism could read in a number of ways, depending on what kind of mathematics you're talking about. You could say Let `f` be a mapping of `A` into `B`, or Let `f` be a mapping of `A` onto `B`, or Let `f` be a function with domain `A` and range `B`. Some prepositions (like into and onto) have meaning by definition; others, like from, invoke some metaphor. In the case of from, it's a motion metaphor, mathematizable as a vector, for instance, or direction of integration. Jun 11, 2019 at 13:43
• @AleksandrH haha, I posted this as a followup question on that very thread, which got deleted by a moderator indicating that this should by asked on a thread of its own.
– Mogu
Jun 11, 2019 at 15:11
• @John Lawler ok, but "into" and "onto" are too specific. I was looking for the English equivalent of the French "sur A dans B". Would (on,in) work?
– Mogu
Jun 11, 2019 at 15:23
• f is a function from A into B. You can also use "mapping" instead of function, and you can use "to" instead of "into". If you really want "defined" then it should be "defined on A", but the combination of that with "into B" or with "to B" sounds awkward to me. Jun 12, 2019 at 2:02

Personally, I would not want to stick more words into the verbalized phrase than there are symbols in the writing. I would prefer to vocalize the given expression as:

"f from A to B"

This might be part of a sentence like, "We define the function f from A to B in the following way...", and so forth.

"Let f be a function from A to B"

if there is nothing else. Variations are on the 'to': 'into'/'onto' for injective/surjective function s respectively (if that is something you want to say there).

If the function is A -> A, you can say more compactly:

"Let f be a function on A."

• I disagree with two things in this answer: "Into" does not mean injective; it's synonymous with "to" in this context. And "function on A" means only that the function's domain is A, not that the codomain is also A; it leaves the codomain entirely unspecified. (In contrast, "unary operation on A" would mean that the domain and codomain are both A. But that's not what this question is about.) Mar 26, 2020 at 2:13