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ECS20
Homework 5
Exercise 1
Find these values :
a ) ë1.1û
b) é1.1ù
d ) é- 0.1ù e) é2.99ù
c ) ë- 0.1û
f ) é- 2.99ù
ê 1 é 1 ùú
éê 1 ú é 1 ù 1 ù
g ) ê + ê úú h) ê ê ú + ê ú + ú
ë 2 ê 2 úû
êë 2 û ê 2 ú 2 ú
Exercise 2 (proof)
a) Show that the following statement is true:
“If x is a real number such that x2+1=0, then x4 = π”.
b) Constructive proof:
“If x and y are real numbers such that x < y, show that there exists a real number z
with x <z < y”
Exercise 3
1ú ê
2ú
ê
Let x be a real number. Show that ë3x û = ëx û + ê x + ú + ê x + ú
3û ë
3û
ë
Exercise 4
Let f be a bijection from a set A to a set B. The inverse of f, noted f -1 is the function that
assigns to an element b in B the unique element a in A such that f(a)=b. Hence f -1(b) = a
when f(a) = b.
Let f be a bijection from a set A to a set B. Let S and T be two subsets of A.
a) Show that
b) Show that
c) Show that
**Extra credit:
Let us consider a generalization of exercise 1. Let x be a real number, and N an integer
greater or equal to 3. Show that:
1ú ê
2ú
N - 1ú
ê
ê
ëNx û = ëx û + ê x + ú + ê x + ú + … + ê x +
Nû ë
Nû
N úû
ë
ë
(Hint: Consider the function f(x) defined as:
1ú ê
2ú
N - 1ú
ê
ê
f ( x ) = ëNx û - ëx û - ê x + ú - ê x + ú - … - ê x +
Nû ë
Nû
N úû
ë
ë
Show first that f(x) is periodic, of period 1/N (i.e. f(x+1/N)=f(x) for all x) )