Chris Fenton
CSF - Discrete Math
HW4
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Section 2.3
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2. Recognize whether f is a function from Z come R if
a) f(n)=±n.

You are watching: The function that assigns to each pair of positive integers the first integer of the pair

No. More than one picture +n and also -n.
b) f(n)=√n^2 +1.
Yes. Since you square n (and add one), friend will always be taking the square source of a confident number.
c) f(n)=1/(n^2−4).
No. No image for n=2 or n=-2.
4. Discover the domain and range of this functions. Keep in mind that in every case, to uncover the domain, determine the set of elements assigned worths by the function.
a) the duty that assigns to every nonnegative integer its critical digit
Domain: 0,1,2,3,...
Range: 0,1,2,3,4,5,6,7,8,9
b) the duty that assigns the next largest integer to a positive integer
Domain: 1,2,3,...
Range: 2,3,4,5...
c) the role that assigns come a bit string the number of one bits in the string
Domain: U0,1 * d, where d is the size of the bit string
Range: 0,1,2,3...
d) the duty that assigns to a little bit string the variety of bits in the string
Domain: U0,1 * d, where d is the size of the bit string
Range: 1,2,3...
6. Find the domain and range of these functions.
a) the role that assigns to every pair of confident integers the very first integer of the pair
Domain: 1,2,3...
Range: 1,2,3...
b) the role that assigns come each positive integer its largest decimal digit
Domain: 1,2,3...
Range: 0,1,2,3,4,5,6,7,8,9
c) the role that assigns to a little bit string the variety of ones minus the number of zeros in the string
Domain: U0,1 * d, where d is the size of the little string
Range: z,...
d) the role that assigns to each optimistic integer the biggest integer not exceeding the square source of the integer
Domain: 1,2,3...
Range: z,...
e) the duty that assigns come a little bit string the longest cable of people in the string
Domain: U0,1 * d, wherein d is the size of the bit string
Range: 0,1,2,...
8. Discover these values
a) ⌊1.1⌋ - 1
b) ⌈1.1⌉ - 2
c) ⌊−0.1⌋ - -1
d) ⌈−0.1⌉ - 0
e) ⌈2.99⌉ - 3
f ) ⌈−2.99⌉ -2
g)⌊1/2 + ⌈1/2⌉⌉ - 1
h) ⌈⌊1/2⌋+⌈1/2⌉+ 1/2⌉ - 2
12. Recognize whether each of these functions from Z come Z is one-to-one.
a) f(n)=n − 1 - Yes
b) f(n)=n^2 + 1 - No, (-n)^2 == n^2
c) f(n) = n^3 - Yes
d) f(n) = ⌈n/2⌉ - No
33. Suppose that g is a duty from A to B and also f is a function from B to C .
a) present that if both f and also g space one-to-one functions, climate f ◦ g is likewise one-to-one.
f(g(x)) = f(g(y))
x = y
40. Allow f it is in a function from the collection A come the collection B. Permit S and also T be subsets that A. Display that
b) f(S∩T)⊆f(S)∩f(T).
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Section 2.4
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4. What space the terms a0, a1, a2, and also a3 the the succession an,
where one equals
(-2)^0 = 1
(-2)^1 = -2
(-2)^2 = 4
(-2)^3 = -8
10. Uncover the an initial six terms of the sequence characterized by each
of these recurrence relations and also initial conditions.
b) one =an−1 −an−2,a0 =2,a1 =−1
2,-1,-3,-2,1,3
c)an=3an2−1,a0=1
1,3,27,2187,14348907,617673396283947
20. Assume the the populace of the civilization in 2010 was 6.9
billion and also is growing at the price of 1.1% a year.
a) set up a recurrence relation because that the populace of the civilization n year after 2010.
(1.011) * P(n-1), P(0) = 6.9
b) discover an explicit formula for the population of the human being n year after 2010.
(1.011)^n * 6.9
c) What will certainly the population of the human being be in 2030?
About 8.6 exchange rate people
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Section 2.5
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2. Identify whether every of these sets is finite, countably infinite, or uncountable. For those that room countably in- finite, exhibit a one-to-one correspondence in between the collection of positive integers and also that set.
a) Infinite. F(x) = 10 + x
b) Infinite. F(x) = -2x - 1
c) Finite
d) Uncountable.
e) f(x) = (2, n) if x = 2n -1
= (3, n) if x = 2n
f) f(x) = 10(n - 1) if x = 2n - 1
= -10n if x = 2n
4. Recognize whether every of these sets is countable or uncountable. For those that room countably infinite, exhibition a one-to-one correspondence in between the collection of hopeful integers and that set.

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a) Countable
b) Countable
c) Countable
d) Uncountable
8. Present that a countably infinite variety of guests getting here at Hilbert’s completely occupied cool Hotel can be offered rooms there is no evicting any current guest.
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