Name:
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Exam 1, Mathematics 109
Dr. Cristian D. Popescu
October t7, 2005
Note: There are 3 problems on this exam. You wili not receive credit unless you
show all your work. No books, calculators,notes or tables are permitted.
I. (30 points)
(1) Write formally the followingstatement: " For all strictly positive real numb e r s r a n d y ,t h e s u mr l A + y l r i s a t l e a s t 2 " .
(2) Write the formalnegationof the statementin (1) above.
(3) Proveor disprovethe statementin (1) above.
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II. (30 points)
(1) Write formally the foilowing statement: "If r is an irrational real number,
then z2 * r is an irrational real number."
(2) Prove or disprovethe statement in (1) above.
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III. (40 points)
(1) Showthat the followingpropositionalexpressionis a tautology.
(P -> Q) ---+((P v Q) <--+ Q)
(2) Usethe tautologyin (1) aboveto showthat if A
a universalsetU suchthat A g B,then A U B :
(3) Proveor disprovethe followingstatement: "If A
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