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# Test_3 - Numbers and Polynomials Test 3 Answer FOUR...

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Numbers and Polynomials Test 3 Answer FOUR questions. Be sure to give reasons for each step. 1. Where possible, in each case give (with justiﬁcation) an example of a nonempty set of real numbers that is bounded above and has: (i) a greatest element; (ii) a least upper bound but no greatest element; (iii) no least upper bound. 2. Let A be a nonempty subset of N that is bounded above (by a real number). Show that its least upper bound lub A is an element of A . 3. Let P : = R \ Q denote the set of all irrational real numbers. For A, B R let A + B : = { a + b : a A, b B } . Identify each of the following subsets of R : (i) Q + Q ; (ii) Q + P ; (iii) P + P . 4. Let p be a prime number and let a, b Q . Show explicitly (and carefully: attend to the denominator) that if a + b p is nonzero then there exist x, y Q such that 1 / ( a + b p ) = x + y p . 5. Let p be a prime number and n a positive integer. Show that if n Q [ p ] and n is not the square of an integer then

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Test_3 - Numbers and Polynomials Test 3 Answer FOUR...

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