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f: x ↦ y means that f is a function which takes in a value x and gives out y. f: N → N means that f is a function which takes a natural number as domain and results in a natural number as the result. Because you're wrong: the → and ↦ arrows mean different things. Also, W is not the set of positive numbers: that's R +.
The modern technical definition of a functional is a function from a vector space into the scalar field. For example, finding the length of a vector is a (non-linear) functional, or taking a vector and returning the 3rd coordinate (relative to some basis) is a (linear) functional. But in a classical sense, functional is an antiquated term for a ...
This notation means that you take the output of h h and use it as the input of f f. When we are working with a specific x x value, we can suggestively write f(h(x)) f (h (x)) instead. (f ∘ h)(x) = f(h(x)) = f(2 + 3x) = 1 2 + 3x. (f ∘ h) (x) = f (h (x)) = f (2 + 3 x) = 1 2 + 3 x. (Note: I only used z z as the variable for f f to avoid ...
783 1 8 14. 4. See Function : "a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output." The "standard" definition start from the set-theoretic def of relation as a set of ordered pairs. – Mauro ALLEGRANZA.
42. Although in most cases the words function and mapping can be used interchangeably, in several parts of mathematics differences in emphasis, especially in analysis and differential geometry. I can think of two. First, especially in differential geometry, "mapping" is the universal word, and the word "function" is used for mappings that map ...
This may not answer the original question, but when I came across this post I was looking for a more mathematical approach (rather than using a defined floor/ceil function). I ended up using modulo to define a floor, ceiling, and "round up from half" Fractions. To floor round after dividing a numerator n by a denominator d:
M (x) is a function. Taking the maximal number amongst the parameters. max{x1,x2} ={x1, if x1>x2 x2, otherwise max {x 1, x 2} = {x 1, if x 1> x 2 x 2, otherwise. You can define like that the maximum of any finitely many elements. When the parameters are an infinite set of values, then it is implied that one of them is maximal (namely that there ...
Nearly $400$ ensuing comments were posted in a debate about what constitutes an operator versus a function. The debate expanded to include a debate on exponentiation (operation vs. function.) I've Googled various sources on the web and the distinction between a mathematical operator vs. a mathematical function seems to be context dependent.
2. To rotate any curve by any angle, you need to use parametric equations. x = t cos θ − f(t) sin θ, y = t sin θ + f(t) cos θ. You get points along the range [s, e] by plugging in values for t starting at s and ending at e. The space between the points is determined by the difference between values of t that you plug in, and each one ...
46. All functions are well-defined; but when we define a function, we don't always know (without doing some work) that our definition really does give us a function. We say the function (or, more precisely, the specification of the function) is 'well-defined' if it does. That is, f: A → B f: A → B is well-defined if for each a ∈ A a ∈ A ...