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MAT102H5 (13)
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# problem set solutions

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Department
Mathematics
Course
MAT102H5
Professor
Shay Fuchs
Semester
Fall

Description
MAT102S - Introduction to Mathematical Proofs - UTM - Spring 2011 Solutions to Selected Problems from Problem Set C 1. 1.17. The domain of y = jxj is R (every number has an absolute value). The image is [0;1) since every y ▯ 0 is the absolute value of itself: y = jyj . 1.35. x=y + y=x ▯ 2 if and only if x and y have the same sign: If x or y is 0, then the expression is unde▯ned. If they have opposite signs, then the left side is negative. If they have the same 2 2 2 2 sign, then multiplying by xy yields x +y ▯ 2xy, equivalent to x ▯2xy+y ▯ 0, equivalent to (x ▯ y) ▯ 0. The last inequality holds whenever x and y have the same sign, so this necessary condition is also su▯cient. 1.49. (a) TRUE. If f and g are bounded, then there exist M an1 M , suc2 that jf(x)j ▯ M 1 and jg(x)j ▯ M 2 for all x 2 R . Using the triangle inequality, we conclude tha
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