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Unformatted text preview: 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 ⇒ Q is equivalent to ( P or Q ) ⇔ Q . Proof. Use truth tables. P Q P ⇒ Q (P or Q) (P or Q) ⇔ Q T T T T T T F F T F F T T T T F F T F T b.(3 points) Show that parenleftbigg a + 1 b parenrightbiggparenleftbigg b + 1 a 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 ≥ which means that x 2 + y 2 ≥ 2 xy (1) In (1), replace x by √ a and y by 1 √ b . We obtain a + 1 b ≥ 2 radicalbigg a b > (2) In (1), replace x by √ b and y by 1 √ a . We obtain b + 1 a ≥ 2 radicalbigg b a > (3) Multiplying (2) and (3), we obtain parenleftbigg a + 1 b parenrightbiggparenleftbigg b + 1 a parenrightbigg ≥ (2 radicalbigg a b )(2 radicalbigg...
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This note was uploaded on 04/30/2008 for the course MATH 109 taught by Professor Knutson during the Winter '06 term at UCSD.
 Winter '06
 Knutson
 Math

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