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Jon Seymore

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In summary, the conversation discusses the properties of inner and outer functions in a composition and how they affect the injectivity and surjectivity of the overall composition. The definition of composition is also discussed, with one participant mentioning Tao's usage of terminology which is not standard. Overall, there seems to be confusion around the terms "domain," "codomain," and "range" and how they relate to each other in defining a function.

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Jon Seymore

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Jon Seymore said:Composing functions by definition requires that the range of the inner function is the domain of the outer function.

This is not true. It suffices that the range of the inner function is a subset of the domain of the outer function. So if ##f:A\rightarrow D## and ##g:B\rightarrow C## are functions and if ##f(A)\subseteq B##, then we define ##g\circ f## with domain ##A## and codomain ##C## to do ##(g\circ f)(a) = g(f(a))##. This is the most general definition of composition. Maybe Tao does not take the definition to this generality.

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Jon Seymore

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Jon Seymore

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Okay, everything is clear to me now. Thank you for your help.

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pbuk

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Jon Seymore

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It seems that Tao defines compositions in a different way. He defines compositions so that the range of f is precisely the domain of g. Maybe he does this because it's more natural from a foundational point of view at this point in his text and maybe it's too restrictive in general;. I'm not sure. Someone else can answer for that better than I can. I just know they're different definitions and I've sorted out how everything works out in each case. I'll tell you how Tao does it specifically. He initializes f and g and then says, "such that the range of f is the same set as the domain of g." That's why in my original comment I was saying things like, "Wouldn't that make f trivially surjective?".MrAnchovy said:

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Jon Seymore said:It seems that Tao defines compositions in a different way. He defines compositions so that the range of f is precisely the domain of g. Maybe he does this because it's more natural from a foundational point of view at this point in his text and maybe it's too restrictive in general;. I'm not sure. Someone else can answer for that better than I can. I just know they're different definitions and I've sorted out how everything works out in each case. I'll tell you how Tao does it specifically. He initializes f and g and then says, "such that the range of f is the same set as the domain of g." That's why in my original comment I was saying things like, "Wouldn't that make f trivially surjective?".

Tao is being highly nonstandard with his terminology here. Usually, one would use the word "codomain" for what he calls "range". So a function has a specific domain and codomain. The range of a function ##f:A\rightarrow B## is then defined as ##f(A)## and may or may not equal the codomain. I don't know why Tao uses the terminology he does, it's really puzzling since everybody does it differently than him.

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Jon Seymore

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Interesting. I thought from the way he seemed to use those words that "codomain" and "range" were the same idea. I guess I'll have to sort through that too. He does in fact use "range" in his definition. This is the first time that I'm hearing those are different ideas. Thanks for letting me know.micromass said:Tao is being highly nonstandard with his terminology here. Usually, one would use the word "codomain" for what he calls "range". So a function has a specific domain and codomain. The range of a function ##f:A\rightarrow B## is then defined as ##f(A)## and may or may not equal the codomain. I don't know why Tao uses the terminology he does, it's really puzzling since everybody does it differently than him.

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Jon Seymore said:Interesting. I thought from the way he seemed to use those words that "codomain" and "range" were the same idea

Yes, he is using them as the same. But in most (modern) texts, they are different. https://en.wikipedia.org/wiki/Range_(mathematics)

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Jon Seymore

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Okay, so I've gathered that compositions are typically defined by initializing functions with codomains in mind, where in general, the range of g, given the range of f as input, is a smaller set than the codomain that we used to initially define g. The codomain of f (by which I mean the initial domain of g) is larger than the range of f. I hope that's all right.micromass said:Yes, he is using them as the same. But in most (modern) texts, they are different. https://en.wikipedia.org/wiki/Range_(mathematics)

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Seems right.

A function's domain is the set of input values for which the function is defined. The codomain is the set of all possible output values of the function.

No, the domain and codomain of a function can be different. The domain can be a subset of the codomain, but it cannot be larger than the codomain.

The range of a function is the set of all output values that the function actually uses. It is a subset of the codomain. The domain and range together determine the function's behavior and how it maps inputs to outputs.

No, a function can only have one domain and one codomain. However, the domain and codomain can be complex sets, such as the set of real numbers or the set of all possible strings.

Defining a function's domain and codomain is important because it helps clarify the behavior and limitations of the function. It also ensures that the function is well-defined and that all inputs will have a corresponding output.

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