MAT102S - Introduction to Mathematical Proofs - UTM - Spring 2010 Solutions to Selected Problems from Problem Set A 1.2. The equation x 2 + bx + c = 0 has exactly one solution when b 2-4 c = 0, and it has no solutions when b 2-4 c < 0. These statements follow from the quadratic formula (in the case where the coeﬃcient of x 2 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 ﬁrst 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 ﬁx 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.
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