Exam I Solutions - Math 347 C1 HOUR EXAM I 29 June 2010...

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Math 347 C1 HOUR EXAM I 29 June 2010 SOLUTIONS 1. Let S = { x R | x 2 > x + 6 } and let T = { x R | x > 3 } . Determine whether the following statements are true. Justify your answers. a) T S . b) S T . SOLUTION Clearly T = (3 , ) . S consists of those real numbers satisfying: x 2 - x - 6 > 0 , and x 2 - x - 6 > 0 if and only if ( x - 3)( x + 2) > 0. This is equivalent to either x - 3 > 0 and x + 2 > 0 OR x - 3 < 0 and x + 2 < 0. In the first case, x > 3 and x > - 2, whence x > 3 . In the second case x < 3 and x < - 2 whence x < - 2 . Therefore S = (3 , ) ( -∞ , - 2) . a) Clearly T S . b) S 6⊆ T. 2. Suppose there are n married couples, each consisting of a man and a woman. How many pairs of a man and a woman are there in which the man and the woman in the same pair are not married? SOLUTION The total number of pairs of men and women is n 2 . Among these are the pairs of a man and his wife; there are exactly n of these. Therefore there are exactly n 2 - n pairs of men and women where no one is paired with their spouse.
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Exam I Solutions - Math 347 C1 HOUR EXAM I 29 June 2010...

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