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How would I write mathematically (using set notation):

Let $F$ be the set defined by the distinct values of the function $f(x)$ for $a \le x \le b$ ($a$, $b$, $x$ and $f(x)$ are all natural integers).

I was thinking of:

$\mathbb{F} = \left \{ \forall x \in \left [a, b \right ] , a, b, x \in \mathbb{N}, f(x) \right \}$

But I'm not sure how to express the fact that the set should only contain distinct values, e.g. if $f(x)$ is equal to say 3, 4, 5, 3, 4 for $a$ through $b$, then $F = \{3, 4, 5\}$.

Can somebody help me out?

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Perhaps it's worth mentioning that the thing you are defining here is called "the image of $[a,b]$ under $f$". – MJD Apr 14 '12 at 14:55
up vote 4 down vote accepted

You probably want $\{f(x); x\in\mathbb N, a\le x\le b\}$.

There is no problem with distinct values, since sets are determined by their elements (membership), so the following two sets are equal: $$\{3,4,5,3,4\}=\{3,4,5\}.$$ Both these notations express the set containing precisely the elements 3, 4 and 5.

See Extensionality and (more advanced) Axiom of extensionality at Wikipedia.

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Excellent! Thanks very much :) I will accept the answer in seven minutes since I have to wait. And I can't vote up either... anyway thanks! – Thomas Apr 14 '12 at 14:46

This is the image of $[a,b]$; it's often written $f([a, b])$, although mathematicians will sometimes shuffle their feet and mumble something about "an abuse of notation". People will be happy for you to write $f([a, b])$ if you also perform this ritual abasement.

Martin Sleziak already pointed out that duplicate values are nothing to worry about, as sets only have a notion of membership, not of multiplicity.

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