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By Arnold, John Charles

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Extra resources for Ego sum Michael: The origin and diffusion of the Christian cult of St. Michael the Archangel (Ph.D., University of Arkansas, 1997)

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For instance, the sine function, as a real CHAPTER 2. ESSENTIAL DICTIONARY 26 function, is not invertible, although it becomes invertible when restricted to the interval [−π /2, π /2]. The following example illustrate the two scenarios. sin−1 ({1}) = π + 2π Z 2 arcsin(1) = π . 2 Things get even more confusing when the reciprocal f (x)−1 of f (x) comes into play (which exists as long as f (x) is non-zero). The quantities f −1 (x) and f (x)−1 are unrelated; for instance, in the appropriate domain, we have sin−1 (x) = arcsin(x) sin(x)−1 = csc(x).

In the case of product, it is advisable to use the full expression to avoid confusion with cartesian product. If X = {x} consists of a single element, then we use the shorthand notation x +Y and xY in place of {x} +Y and {x}Y , respectively. For example 1 3 5 1 +N = , , ,... 2 2 2 2 3Z = {. , −6, −3, 0, 3, 6, . }. This notation is economical and effective; it leads to concise statements such as mZ + nZ = gcd(m, n)Z. ) Elementary —but significant— applications of this notation are found in modular arithmetic.

ESSENTIAL DICTIONARY 28 We have merely replaced the solution set of an equation with the equation itself. This identification provides the desired bi-unique correspondence between the two sets. We can simplify further. Because a and b are given, there is no need to specify them explicitly: it suffices to give the (possibly infinite) value of the slope. Alternatively, we could identify a line by an angle between 0 and π , measured with respect to some reference axis passing through the point (a, b).

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