Fall 2005 - Popescu's Class - Exam 1

# Fall 2005 - Popescu's Class - Exam 1 - Exam 1 Mathematics...

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Exam 1, Mathematics 109 Dr. Cristian D. Popescu October t7, 2005 Name: Student ID: Section Number: 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 following statement: " For all strictly positive real num- bersr andy, thesum rlA+ylris atleast2". (2) Write the formal negation of the statement in (1) above. (3) Prove or disprove the statement x'l€ {R' t,) [v -) ("r) (e it 7),r tXt R.r.) > frt GI Q-)trr) (" ^ (b*|* 42) y rovt il* Sf^+c *a^+ l'n ( ^ ) ,l e R>o L +)- -z = TX %'^tt x ,le Q>o ) A \&^V q-:j' >o -tr^F S/t wi\\ Ulx Tti^ erRro)^.hQIRrJ XJ -f >" L+I \x Y +I T1 x'* f- zx Tt-*t- D

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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 disprove the statement in (1) above. c) t (\ \-- I 1(-Q R T* qf^k"^l-A \n tn) )\ tJ-* ' l'Je *;l\ q(^^ ^ {u^', \a T"-'.,\ lt'^* if s tr'-t3<'f'nt \ q \-" \^ '^"'lt;* + S"{ s+^J<'^ r'+ a'L'"-'re is Q,.)G4q)^3'+x€G) \2- 5n-= w\ "-\ q\ *- --) \* -\ --) ] \-eZ ) an = bqA . }"^"F^
T--"tr 5 n..- ?'R' ( \ a \o ^^)- f*,, - \.-j,^^*\*\ \-= [4'J -\)**ts '"'r* c) 1 -:=) s\ n- ;> 1'dL*',

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Fall 2005 - Popescu's Class - Exam 1 - Exam 1 Mathematics...

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