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11a-functions - The Mathematics of Functions The-calculus...

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The Mathematics of Functions The λ -calculus is very low level. Here we investigate functions in a more abstract setting. This is very similar to writing algorithms in English prose rather than writing actual machine code for the Turing machines. – p. 1/
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The Mathematics of Functions Function Application and Composition: Let f : A B be a (total) function from A to B , then for every value x A we obtain a value y B , fx = y Function application is expressed by the juxtaposition of the function and its argument and is evaluated from right to left. Now assume that we have another function g : B C from B into C , then we can apply the function g to the result of f . For every value in x A we obtain a value z C , gfx = gy = z In other words, we just constructed a new function, call it h : A C , such that hx = gfx = gy = z We can express the same idea using function composition, , without having to explicitly reference any values in A , B , or C , h = g f and we say that “ h
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