sols11

sols11 - 11. OCTOBER 6TH: FUNCTIONS: EXAMPLES. COMPOSITION...

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11. OCTOBER 6TH: FUNCTIONS: EXAMPLES. COMPOSITION OF FUNCTIONS. 27 11. October 6th: Functions: examples. Composition of functions. 11.1 . Let A denote a set { F,T } with exactly two elements, and let S be any set. Prove that giving a function S A (that is, an element of A S )is‘thesameas ’g iv ing asubseto f S (that is, an element of P ( S )). In other words, explain how to deFne a subset of S for every function S →{ F,T } ,andhowtodeFneafunct ion S →{ F,T } for every subset of S ,incompa t ib leway s . (Th i si swhy P ( S )i sa l sodeno ted2 S ; here, ‘2’ stands for a set with two elements.) Answer: Given a function f : S →{ F,T } ,let X f S be the subset deFned by X f = f 1 ( T )= { s S | f ( s )= T } . In this way, every function S →{ F,T } determines a subset of S . Conversely, given a subset X S of S ,deFneafunc t ion f X : S →{ F,T } by prescribing s S f X ( s )= °
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.

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sols11 - 11. OCTOBER 6TH: FUNCTIONS: EXAMPLES. COMPOSITION...

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