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Unformatted text preview: I. { 5 Pts) State the Fundamental Theorem of Arithmetic. ,2 ,2“ .2
II. ( 10 Pts) Determine the number of divisors of 12,100. 4:“ N " 2 v 5 III. ( 10 Pts) Use the Fundamental Theorem ofArithrnetie to prove that the square root of 300 is
irrational IV. ( 8 pts) Prove that if A is a nonzero rational and B is irrational, then BIA is irrational. V. ( 5 Pts) Determine the fraction corresponding to the base two expression. .101 T. 2:4; VI. ( 24 Pts) Express each of the following in the form x + yi where x and y are real numbers. {1) (4 31 y {1 —2i) 2 + '5 (2). (4441305: faking : 32 Moi311:: 32 Liza‘qu = dig—maﬁa i w :' ‘L
(3). The three cube roots of *Si. *5 8 @5391; g: “ l “A A ’2’
EZ’ZLéhig :1 —lr§“’,&
VII. {8 Pts) Find t?enegationof(p A q) —>(q_——>r), 23 :2 53,, 331: = +5 a”:
P 1‘ ﬁ’ ) A f g, A «— we}
VIII. ( 10 Pts) State the contrapositive of the given statement and use this contrapositive to prove
or disprove the given statement. / pf : m = 2 + i 34 “$2.3 “ If (to +702 + 2n is not divisible by 4, then m is even or n is odd. “ (3/1 Mile 2 n = _ WWmeszf; (m+r}?+2n Wis3%
IX; ( 12 Pts) Determine if the following relation deﬁned on the set of integers is: (a). R@ (b)« 5%0 (o). TEEEve.
o Na ny C> XZny 20. Kerwin? 4 1(23) " X. ( 12 Pts) Let ﬁx) be the sawtooth function with 5 teeth (1). Determine the number of elements in the orbit of 3327 2574 ~ Q
(2). Find 3 different limit points for ‘23233233323333233333;.. . 3 6% ’ L W (3). Give an example of a number within .00001 of 1/3 whose orbit ends in a 3cyclet
, 323 3333 25:6 XI.( 8 Pts) Let f(x) be the tent function. (1) Express f 5 (2/9) as a base 2 expression. n l i i O 0 O
‘ 1 9 D l O l l l (2) Determine all points of period 3 for the tent function. E r 733’ .W
‘ . u o . o o; t a 0 XII. ( 10 Pts) In the following determine the ﬁxed points of the given function and whether each
of these is attracting, repelling or neither attracting nor repelling.
‘ e m f(x)=x3+(9f25)x. 3 wt r
“TM XIII. (16 Pts) Give an example of each of the following, if one exists. lfno example exists,
write “None”. i Y t: F j (5‘ (1). Five different capital letters that are topologmally equivalent to i . NEW—«d , 0400:0061“.. . . . .
g g o H o t E ‘ o M h (2). Two irrationals whose sum is rational. Q r 3» l (3) A value of “a” for which the function f(x) == ax(l — x) has an attracting 2—cycle. N :29, 929g; (4) Three different perfect numbers. XIV. ( 12 Pts) Answer True or False. «7” (l). The set of all rational numbers has the same cardinality as the set of
' all algebraic numbers. F7 (2). If U : the set of rationals, then the following quantiﬁed statement is true: “(veeyx x + f = 0)”° F" (3). For the order relation on the natural numbers given by xsg y if and only if
x divides y, the least upper bound of the set A = { 30, 45, 300} is 600. “T” (4) {0,112} is an attracting Zscylee for the function f(x) 2 2x2 —2x + 1/2. ...
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This note was uploaded on 05/31/2008 for the course MATH 2600 taught by Professor Sprows during the Spring '08 term at Villanova.
 Spring '08
 Sprows
 Math

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