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Winter 2008 - Cioaba's Class - Exam 1 (Version 2)

Winter 2008 - Cioaba's Class - Exam 1 (Version 2) - Name...

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Name: PID: Midterm 1, Math 109 - Winter 2008 Duration: 50 minutes Please close your books, turn off your calculators and phones. To get full credit you should explain your answers. 1.a. (2 points) Show that ( P and Q ) P is equivalent to P Q . Proof. Use truth tables. P Q P Q (P and Q) (P and Q) P T T T T T T F F F F F T T F T F F T F T b.(3 points) Show that ( a + b ) parenleftbigg 1 a + 1 b parenrightbigg 4 for any real and positive numbers a and b . When does equality hold ? Proof. Solution 1 If x and y are real numbers, then ( x - y ) 2 0. This implies that x 2 + y 2 - 2 xy 0 which means that x 2 + y 2 2 xy (1) In (1), replace x by a and y by b . We obtain a + b 2 ab > 0 (2) In (1), replace x by 1 a and y by 1 b . We obtain 1 a + 1 b 2 1 ab > 0 (3) Multiplying (2) and (3), we obtain ( a + b ) parenleftbigg 1 a + 1 b parenrightbigg (2 ab )(2 1 ab ) = 4 . Equality happens iff equality happens in (2) and in (3). This happens if and only if a = b and 1 a = 1 b which is equivalent to a = b .
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