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

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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 ‘the same as’ giving a subset of S (that is, an element of P ( S )). In other words, explain how to define a subset of S for every function S { F, T } , and how to define a function S { F, T } for every subset of S , in compatible ways. (This is why P ( S ) is also denoted 2 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 defined 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 , define a function f X : S { F, T } by prescribing s S f X ( s ) = T if s X F if s X . In this way, every subset of S determines a function S { F, T } .
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