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CS173: Discrete Mathematical Structures
Spring 2006
Homework
Due 02/19/
Solutions (35 points)
- (Points given on compass independently) For this problem, please log in to compass and answer the 5 true/false questions contained in Quiz 1.
Solutions: 2 n^ = o( n^10000 ) False. 2 n^ = O(n! )True. n ร log( n)=o( n^1.^0000000001 )True. n!=little-omega(n^(10^(10^100)) True. 1 = O(2^10000) True.
- (12 points), 3 points for each correct question, -1 point for bad domain, bad big-O or any other partial error. โ2 for major error
Consider the four following functions: f (x) = lne (lne (x)) , g(x) = ex^ , h(x) = x^5 ,
t(x) = x^10. Find the domain of definition and the big-O of the following functions: a. ( f o g o h)(x) Solution: ( f o goh)(x)= 5 ln(x). Thus, the domain is x>0, and O(ln(x)) b. ( f โ^1 o g)(x)
Solution: eex ( f โ^1 o g)(x)= e. Thus, the domain is all real numbers,
and ( ) eex O e c. (gโ^1 o h)(x) Solution: ( g โ^1 o h)(x)= 5 ln(x). Thus, the domain is x>0, and O(ln(x))
d. g(x) *[( f o tโ^1 )(x)]
Solution: ) ln(ln( ))] 10
g ( x)ร [(f o tโ^1 )(x)]=ex^ [ln( + x. Thus, the
domain is x>0, and O( ex^ ln(ln(x)))
- (12 points) 2 points for each correct answer.
a. Evaluate:
( j + 1)^2 j = 2 (2i^ โ^ 1)
5 โi
i = 1
4
Solution: ( 3 2 / 1 + 42 / 1 + 52 / 1 )+( 32 / 3 + 42 / 3 )+( 32 / 5 )= 60. 1
b. Let f(x) be the set of all even numbers smaller or equal than x. For example, f(10)={0,2,4,6,8,10}. Let g(x)=2x.
Compute: j j โ( f og )(i )
i = 1
4
Solution: 2+(2+4)+(2+4+6)+(2+4+6+8)=
c. There is also a special notation for products: an i = 1
n
โ =^ a 1 *^ a 2 ...^ an.
Compute: (โ 1 )i^2
i = 5
10
โ =^
Solution:
(^ โ 1 )i^2
i = 5
10
( โ 1 )^5 2 ร(โ 1 )^62 ร(โ 1 )^72 ร(โ 1 )^82 ร(โ 1 )^92 ร(โ 1 )^102 =โ 26
d. Express this sum in terms of n:
j
j + 1
j = 1 ๏ฃธ๏ฃท
n
(1-1/2)+(1/2-1/3)+โฆ([1/n]-[1/(n+1)])= 1+(-1/2+1/2)+(-1/3+1/3)+โฆ (-[1/n] +[1/n])-1/(n+1)= 1-1/(n+1)
e. Express this sum in terms of n: i i = 1
k
k = 1
n
1 1 1
2 1 1
โโ =^ โ + = โ +โ
= = = = =
i kk k k nn n n n
n
k
n
k
n
k
n
k
k
i
f. Express n! using product (โ ) notation.
=
n
i
i 1
Case 2: Both A and B-A are countably infinite. There are bijective functions f: AN and g: (B-A) N Define a function h:NAUB as follows: h(x) = f(x/2) when x is even, g((x+1)/2) when x is odd
