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). | |

============= | |

Section 2.4 | |

============= | |

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. See more: Mary J Blige You Dont Have To Worry, You Don'T Have To Worry (Mary J | |

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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