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MAT102H5 (13)

problem set solutions

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Shay Fuchs

MAT102S - Introduction to Mathematical Proofs - UTM - Spring 2011 Solutions to Selected Problems from Problem Set A 1.2. The equation x +bx+c = 0 has exactly one solution when b ▯4c = 0, and it has no solutions when 2 b ▯4c < 0. These statements follow from the quadratic formula (in the case where the coe▯cient of x is a = 1). 1.7. If we take x = 2 and y = ▯1, then x > y but ▯1=x = ▯1=2 is NOT greater than ▯1=y = 1. If we replace the condition x > y by x > y > 0 then we get: x > y ) 1=x < 1=y ) ▯1=x > ▯1=y as required (in the ▯rst step we divided both sides by xy, and in the second step we multiplied both sides by (▯1)). Note: This is not the only way to ▯x the false statement. 1.23. A digital 12{hour clock broken so that the readings for minutes and for hours are always the same can be correct every 61 minutes, except that between 12:12 and 1:01 there are only 49 minutes. Therefore the minimum number of minutes is 49. 1.25. We assume that the ages are positive integers. Let them be a;b;c with a ▯ b ▯ c. We are told that abc = 36, but that knowing a
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