Definition: Function, Function Value, Fiber, Image and Inverse Imagedeleteeditadd to favorites[id:592]vote: last edited 1 month ago by bookofproofs Notation show notationLet \(A\) and \(B\) be sets. \(f\) is a function from the set \(A\) to the set \(B\) (shortly \(f:A\mapsto B\)) if, and only if:
The element \(b\), which is uniquely determined by the element \(a\), is called the function value at \(a\) or the image of \(a\) under \(f\) and denoted by \(f(a)\). In this case the equation \[f(a)=b\] holds. The set \[f(A):=\{f(x)\in B~|~x\in A\}\subset B\] is called the image of \(A\) under \(f\). The set \[f^{-1}(B):=\{x\in A~|~f(x)\in B\}\subset A\]is called the inverse image of \(B\) under \(f\). In general, the set \[f^{-1}(b):=\{x\in A~|~f(x)=b\in B\}\subset A\] can contain more then one element in \(A\), and is called the fiber of b under \(f\). What follows from what?This is (experimental) work in progress - if you miss an axiom, a definition, a theorem or a proof, if you find any inconsistencies you want to correct, or just know about a cool example or explanation you want to share with others, then join our team and help to improve this catalogue. Learn more about the axiomatic approach on BoP...Contribute to BoP: add a new Motivation add add a new Example add add a new Application add add a new Explanation add add a new Interpretation add add a new Corollary add add a new Algorithm add add a new Bibliography (Branch) add add a new Definition add add a new Comment (Branch) add |
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