The Mathematical definition of a function includes the codomain as part of the definition. Source: https://en.wikipedia.org/wiki/Function_(mathematics) …, and I've just checked Halmos' "Naive Set Theory" as the nearest authoritative book which agrees with the definition given on Wikipedia.
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I agree! I’ve been arguing for this position for years. But I still want to know what the people think. Frankly we are losing the battle of ideas. I am unable to convince my students and colleagues. Perhaps 1 out of 100 U.S. high school math teachers teach the codomain this way.
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What practical benefit do you get by calling them different? Why not call them "the same" and disambiguate what you mean by "same" if/when it matters?
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I would say: it depends on the mathematical area your working in and/or on the context. Always use the definition that's the more practical in your situation!
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Two examples: (1) If in your context, surjectivity matters, then you'd rather consider that the codomain is part of the definition.

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Hvala. Twitter će to iskoristiti za poboljšanje vaše vremenske crte. PoništiPoništi
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My discrete math textbook says that function is a set of pairs (x, y), and they are same if and only if two sets are equal to each. Therefore since f and g here have the same sets, they are same functions.
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That definition is limited.
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when i’m reading other people’s work, i interpret equality of functions being defined by having equal domains and codomains, and the images of their domains being equal. this is the only way we could have a notion of a function being intrinsically surjective.
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...personally i feel like for a given domain the only codomain that really matters is the image.
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Čini se da učitavanje traje već neko vrijeme.
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