In the function $f:\mathbb{N}\rightarrow\mathbb{N}$ defined by $f(x)= 5x$ why is range not equal to co domain? I don't understand why this is not a surjective function while the same function is surjective if $f:\mathbb{R}\rightarrow\mathbb{R}$.
Question concerning the domain of a function
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special-functions
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3What's with the title? Could you explain a motivation? – 2017-02-15
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1If by "range" you mean "image" then the image is the multiples of five. For example, $1$ is not in the image. In this context the "codomain" would still be $\mathbb N$. Whether or not the function is surjective on the reals has nothing to do with it. – 2017-02-15
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1I like pizza, but only when it is relevant to the question. – 2017-02-15
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0For example: if $f$ were surjective, there must be an $n$ from the domain such that $f(n) = 1$ – 2017-02-15
1 Answers
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If $f$ is defined on the natural numbers you have $im(f)=5\mathbb{N}=\{5,10,15,20,\dots\}$. If $f$ is defined on $\mathbb{R}$ then given any $x\in \mathbb{R}$ there is a $y:=\frac{x}{5}\in\mathbb{R}$ such that $f(y)=x$. Hence $f$ is surjective.