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If we use the division methods with m = 2^r for some integer r. Why would...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
In the following problem, we will work through a proof of an important theorem of arithmetic. Your job will be to read the proof carefully and answer some questions about the argument. Theorem (The Di...
In the following problem, we will work through a proof of an
important theorem of arithmetic. Your job will be to read the proof
carefully and answer some questions about the argument. Theorem
(The Division Algorithm). For any integer n ≥ 0, and for any
positive integer m, there exist integers d and r such that n = dm +
r and 0 ≤ r < m. Proof: (By strong induction on the variable n.)
Let m be an arbitrary...
Would you agree to the statement on random sampling methods. Please explain why. In probability samples...
Would you agree to the statement on random sampling methods. Please explain why. In probability samples “each population element has a known (non-zero) chance of being chosen for the sample.” (StatTrek 2020). Some examples of probability samples are, simple random sampling, stratified sampling, cluster sampling, multistage sampling, and systematic random sampling. Simple random sampling is the population and sample consists of “N” objects, and an example is when people play the lottery. Stratified sampling is based on some type of...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we ...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
(a) Design an algorithm that reveals some secret integer number from the set {1, 2, ......
(a) Design an algorithm that reveals some secret integer number from the set {1, 2, ... , n} by guessing random numbers within the given range until the secret number has been guessed. Numbers should be replaced after being guessed such that it is possible to guess 2 and then 2 again, assuming 2 is in the given range. The algorithm should return both the secret number as well as the number of guesses taken. (b) If possible, calculate the...
Please help thanks! 2. We now use the idea of revealed preference to consider some of...
Please help thanks!
2. We now use the idea of revealed preference to consider some of the issues that arise in evaluating the effects of inflation. To keep things simple, we first work with the case of two goods, and hence a person who maximizes the function U21,22) subject to the budget constraint pıtı + P202 = M. Our measure of "better off" and "worse off" will be "higher utility" and "lower utility". 2.1 First, suppose that Pı and p2...
From the proof of (ii) . Explain/Show why -n+ 1Sm-kn-1 is true by construction. . Explain/Show wh...
From the proof of (ii) . Explain/Show why -n+ 1Sm-kn-1 is true by construction. . Explain/Show why 0 is the only number divisible by n in the range -n+1 ton-1 Proposition 6.24. Fix a modulus nEN. (i) is an equivalence relation on Z. (ii) The equivalence relation-has exactly n distinct equivalence classes, namely (ii) We need to prove that every integer falls into one of the equivalence classes [0], [1],..., [n -1], and that they are all distinct. For each...
(2) (4 pts) We are going to generalize the result from the previous exercise as follows. Fix two ...
(2) (4 pts) We are going to generalize the result from the previous exercise as follows. Fix two positive numbers c and c2 satisfying cI <c. Define number of primes between ciN and c2N number of integers between ciN and cN This is the probability that an integer n in the interval cN,cN s a prime number. Use the Prime Number Theorem to find an easy to compute function F(c,c2; N) such that P(C1,C2; N) lim (3) (3 pts each)...
1. State and explain the definition of big-O. 2. Explain why we use big-O to compare...
1. State and explain the definition of big-O. 2. Explain why we use big-O to compare algorithms. 3. Explain why binary search runs in O(log n) time. 4. Under what conditions is it possible to sort a list in less than O(nlog n) time? 5. List and explain the worst-case and average-case running times for each Vector method below: (a) insert(iterator here, Object item) (b) insertAtHead (c) insertAtTail (aka push back) (d) get(iterator here) (e) get(index i) (f) remove(iterator here)...
We know that we can reduce the base of an exponent modulo m: a(a mod m)k...
We know that we can reduce the base of an exponent modulo m: a(a mod m)k (mod m). But the same is not true of the exponent itself! That is, we cannot write aa mod m (mod m). This is easily seen to be false in general. Consider, for instance, that 210 mod 3 1 but 210 mod 3 mod 3 21 mod 3-2. The correct law for the exponent is more subtle. We will prove it in steps (a)...
1. [10 marks] Modular Arithmetic. The Quotient-Remainder theorem states that given any integer n and a positive integer d there exist unique integers q and r such that n = dq + r and 0 r< d. We define the mod function as follows: (, r r>n = qd+r^0<r< d) Vn,d E Z d0 Z n mod d That is, n mod d is the remainder of n after division by d (a) Translate the following statement into predicate logic:...
In the following problem, we will work through a proof of an
important theorem of arithmetic. Your job will be to read the proof
carefully and answer some questions about the argument. Theorem
(The Division Algorithm). For any integer n ≥ 0, and for any
positive integer m, there exist integers d and r such that n = dm +
r and 0 ≤ r < m. Proof: (By strong induction on the variable n.)
Let m be an arbitrary...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Please help thanks!
2. We now use the idea of revealed preference to consider some of the issues that arise in evaluating the effects of inflation. To keep things simple, we first work with the case of two goods, and hence a person who maximizes the function U21,22) subject to the budget constraint pıtı + P202 = M. Our measure of "better off" and "worse off" will be "higher utility" and "lower utility". 2.1 First, suppose that Pı and p2...
From the proof of (ii) . Explain/Show why -n+ 1Sm-kn-1 is true by construction. . Explain/Show why 0 is the only number divisible by n in the range -n+1 ton-1 Proposition 6.24. Fix a modulus nEN. (i) is an equivalence relation on Z. (ii) The equivalence relation-has exactly n distinct equivalence classes, namely (ii) We need to prove that every integer falls into one of the equivalence classes [0], [1],..., [n -1], and that they are all distinct. For each...
(2) (4 pts) We are going to generalize the result from the previous exercise as follows. Fix two positive numbers c and c2 satisfying cI <c. Define number of primes between ciN and c2N number of integers between ciN and cN This is the probability that an integer n in the interval cN,cN s a prime number. Use the Prime Number Theorem to find an easy to compute function F(c,c2; N) such that P(C1,C2; N) lim (3) (3 pts each)...
We know that we can reduce the base of an exponent modulo m: a(a mod m)k (mod m). But the same is not true of the exponent itself! That is, we cannot write aa mod m (mod m). This is easily seen to be false in general. Consider, for instance, that 210 mod 3 1 but 210 mod 3 mod 3 21 mod 3-2. The correct law for the exponent is more subtle. We will prove it in steps (a)...
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