I am having problems breaking this down in order for it to make sense in my head. How would you read this? $f(S)=\{f(x)\,|\,x \text{ exists in }S\}$
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Where did you find this? – Lord Soth Jul 03 '13 at 03:18
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From my textbook. S represents a set. – user84833 Jul 03 '13 at 03:21
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1Ok now we are getting somewhere with the edit. – Lord Soth Jul 03 '13 at 03:24
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x is a subset of S? – Elliot Bonneville Jul 03 '13 at 03:25
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2I think you need one more edit: $f(\mathcal{S}) = {f(x)|x\mbox{ exists in }\mathcal{S}}$. Would you double-check? – Lord Soth Jul 03 '13 at 03:25
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Are you sure its not meant to read, "$x$ is an element of $S$"? Btw, you can put dollar signs around math to make it look mathy. Within the dollar signs, write { } for open and close braces, and \in for the membership symbol. – goblin GONE Jul 03 '13 at 03:26
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What does "x exists in $S$" mean? Is that really what the book says? By the way: What textbook is this? "My textbook" is not really helpful. – Andrés E. Caicedo Jul 03 '13 at 03:26
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1@AndresCaicedo It sounds more romantic than $x\in\mathcal{S}$. – Lord Soth Jul 03 '13 at 03:27
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Given a set $S$ and a function $f$, it is customary to define $$ f(S) = \{f(x) \mid x \in S\}. $$ With this definition, $f(S)$ is the image of $S$ under $f$. In other words, you can obtain $f(S)$ by plugging each element of $S$ into $f$ and collecting the outputs.
Austin Mohr
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The assertion that $f(S)=\{f(x) \mid x \in S\}$ amounts to the following.
- Whenever you see the expression $x=f(s),$ if you furthermore know that $s \in S$, then you may deduce that $x \in f(S)$.
- Whenever you see the expression $x \in f(S)$, you may deduce that there exists $s \in S$ such that $x = f(s)$.
goblin GONE
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