MAT102S - Introduction to Mathematical Proofs - UTM - Spring 2010Solutions to Selected Problems from Problem Set A1.2.The equationx2+bx+c= 0 has exactly one solution whenb2-4c= 0, and it has no solutions whenb2-4c <0. These statements follow from the quadratic formula (in the case where the coefficient ofx2isa= 1).1.7.If we takex= 2 andy=-1, thenx > ybut-1/x=-1/2 is NOT greater than-1/y= 1.If we replace the conditionx > ybyx > y >0 then we get:x > y⇒1/x <1/y⇒-1/x >-1/yas required (in the first step we divided both sides byxy, and in the second step we multiplied bothsides 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 samecan 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 bea, b, cwitha≤b≤c. We are told thatabc= 36, but that knowinga+b+c
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