Proofs: Algebra of Composition of Functions
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3. This problem is about the algebra of composition of functions:
a) Let f: A--> B, g: B --> C, and h: C --> D be functions. Show that (hog)of = ho(gof).
b) Let f: A -->B be a function. Show that I(subscript B)of = f = foI(subscript A)
c) Let A = {a, b} be a set with two elements. Find functions f, g: A--> A such that fog does not equal gof.
d) Suppose that A is a non-empty set and fog = gof for all functions f, g: A -->A. Show that A has one element.
e) Let f : A --> B and g, h B -->A be functions. Suppose that gof= I(subscript A) and foh = I(subscript B). Show that h = g.
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This solution is enclosed within an attached Word document and is presented in a step by step manner. All mathematical symbols are represented appropriately in the solution file.
Education
- BSc , Wuhan Univ. China
- MA, Shandong Univ.
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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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