Let $P = \{ 1,2,3 \}$ and $I = \{2,4\}$. And $f:P\to I$ such that $f(1) = 2$ and $f(2) =4$.
Does it make sense in this case to say that $f$ is surjective or injective even if $f$ is not defined on $3 \in P$?
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3$\begingroup$ Since $f$ is only defined at $1$ and at $2$, writing $f\colon\{1,2,3\}\longrightarrow\{2,4\}$ makes no sense. $\endgroup$– José Carlos SantosJan 19, 2022 at 13:40
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3$\begingroup$ $f$ is not a function on $P$. $\endgroup$– RandallJan 19, 2022 at 13:41
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1$\begingroup$ @Randall@JoséCarlosSantos Actually in my course we are studyig in french, and we call such a thing an 'Application'. And we treat function as a special case of Applications. So Applications are injective and surjective. $\endgroup$– mmmmh mmmmmhJan 19, 2022 at 13:49
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What you are talking about are called partial functions. And yes, you can talk about injectivity and surjectivity with them however, you have to restrict to the domain where they become functions. So you don't really gain anything.
