Let $f: X \longrightarrow Y$ a map, if $\:\operatorname{Dom}(f)$ is a countable set, show that $\operatorname{Ran}(f)$ is a countable set.
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What is $X$? And $Y$? $f$ is any function? – azif00 Sep 24 '19 at 23:53
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$X$ and $Y$ any sets and $f$ is any function – Danko1511 Sep 24 '19 at 23:56
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I am certain there are better duplicate targets. But this one will have to do. – Asaf Karagila Sep 25 '19 at 07:15
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$f$ is surjective onto its range. Whenever you have a surjection from $X$ to $Y$, it follows that $\mid Y\mid\le\mid X\mid$. (This is a basic fact from set theory.)
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Not so basic. It uses the axiom of choice, which is unnecessary to this exercise. – Asaf Karagila Sep 25 '19 at 07:10