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Problem 8: 10 points. Prove the inclusion-exclusion theorem for n = 3, i.e., prove that for...
Problem 5, 10 points Roll three (6-sided) dice. Use inclusion-exclusion to find the probability that at...
Problem 5, 10 points Roll three (6-sided) dice. Use inclusion-exclusion to find the probability that at least one value of "2" appears. Hint: Consider A, to be the event that the ith dice shows a "2" for i 1,2,3. We want to find P(A1 UA2U A3) using PI.E. for 3 events. You can assume that each dice is fair, that is, P(A) 1/6, P(Ai n A) 1/6x 1/6-1/36 and P(An A2nA3) (1/6)3 1/216. For an easier solution, consider the complement...
Please do only Problem 4! Use 3 as result. 3. Use the inclusion-exclusion formula derived in...
Please do only Problem 4! Use 3 as result.
3. Use the inclusion-exclusion formula derived in class as well as induction on the integer n to show that for any sequence of events {AjlI, we have that j-1 This upper bound is referred to as the union bound. 4. Extend the above result to show that we have the analogous bound P( A) P(A), j-1 for the case of an arbitrary, but countable, number of events } Hint: Use the...
3. [10 points] Consider the following theorem. Theorem. Assume that m is an integer that leaves...
3. [10 points] Consider the following theorem. Theorem. Assume that m is an integer that leaves a remainder of 6 upon division by 8. Assume furthermore that n is an integer that leaves a remainder of 3 upon division by 8. Then the product m n leaves a remainder of 2 upon division by 8. Consider the tollowing theorern. (a) Illustrate the theorem using an example. (b) Prove the theorem.
Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5)
Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5)
Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5)
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
3. (a) Prove the following: Cantor's Intersection Theorem: Let (X, d) be a complete metric space...
3. (a) Prove the following: Cantor's Intersection Theorem: Let (X, d) be a complete metric space and {Anymore a nested sequence of non-empty closed sets whose diameters D(An) have limit 0. Then An has exactly one member. csc'anno proach onsdelered. c) Show that, in part (a), n A, may be empty if the requirement that the diameters
Problem 1. (4 pts) Combinatorics and the Principle of Inclusion Exclusion (a) (2pts) Roll a fair...
Problem 1. (4 pts) Combinatorics and the Principle of Inclusion Exclusion (a) (2pts) Roll a fair die 10 times. Call a number in 1, 2, 3, 4, 5, 6 a loner if it is rolled exactly once on the 10 rolls. (For example, if the rolls are 1 2 6 4 4 4 6 3 4 1, then 2 and 3 are the only loners) Compute the probability that at least one of numbers 1, 2, 3 is a loner....
3. (a) Prove the second part of the theorem from the (Sept. 23) notes: Theorem Let...
3. (a) Prove the second part of the theorem from the (Sept. 23) notes: Theorem Let A = Mmxn : RM → RM. Then R(A)+ = N(AT). Hint: Take an arbitrary xEN (AT). (b) Verify both sides of the relation above for the matrix A= [ 1 11 21 -1 0 2 -3 . 1 2 4 8 2 |
Problem 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red...
Problem 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of...
Prove that mutual exclusion is satisfied by the following algorithm for the critical section problem: (image...
Prove that mutual exclusion is satisfied by the following
algorithm for the critical section problem:
(image attached)
Note that the if statement is atomic.
By "if statement is atomic" it means that lines p2 and q2 are
either executed in their entirety once execution begins. No
interleaving is possible for example, between the "if" test and the
assignment of wantp/wantq.
You must prove the appropriate program invariant with the help of
induction. It's okay to prove lemmas through program
invariant-induction...
Problem 5, 10 points Roll three (6-sided) dice. Use inclusion-exclusion to find the probability that at least one value of "2" appears. Hint: Consider A, to be the event that the ith dice shows a "2" for i 1,2,3. We want to find P(A1 UA2U A3) using PI.E. for 3 events. You can assume that each dice is fair, that is, P(A) 1/6, P(Ai n A) 1/6x 1/6-1/36 and P(An A2nA3) (1/6)3 1/216. For an easier solution, consider the complement...
Please do only Problem 4! Use 3 as result.
3. Use the inclusion-exclusion formula derived in class as well as induction on the integer n to show that for any sequence of events {AjlI, we have that j-1 This upper bound is referred to as the union bound. 4. Extend the above result to show that we have the analogous bound P( A) P(A), j-1 for the case of an arbitrary, but countable, number of events } Hint: Use the...
3. [10 points] Consider the following theorem. Theorem. Assume that m is an integer that leaves a remainder of 6 upon division by 8. Assume furthermore that n is an integer that leaves a remainder of 3 upon division by 8. Then the product m n leaves a remainder of 2 upon division by 8. Consider the tollowing theorern. (a) Illustrate the theorem using an example. (b) Prove the theorem.
Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5)
Problem (2), 10 points Let n be an integer. Prove that if 3 does not divide n, then 3(2n2 5)
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
3. (a) Prove the following: Cantor's Intersection Theorem: Let (X, d) be a complete metric space and {Anymore a nested sequence of non-empty closed sets whose diameters D(An) have limit 0. Then An has exactly one member. csc'anno proach onsdelered. c) Show that, in part (a), n A, may be empty if the requirement that the diameters
Problem 1. (4 pts) Combinatorics and the Principle of Inclusion Exclusion (a) (2pts) Roll a fair die 10 times. Call a number in 1, 2, 3, 4, 5, 6 a loner if it is rolled exactly once on the 10 rolls. (For example, if the rolls are 1 2 6 4 4 4 6 3 4 1, then 2 and 3 are the only loners) Compute the probability that at least one of numbers 1, 2, 3 is a loner....
3. (a) Prove the second part of the theorem from the (Sept. 23) notes: Theorem Let A = Mmxn : RM → RM. Then R(A)+ = N(AT). Hint: Take an arbitrary xEN (AT). (b) Verify both sides of the relation above for the matrix A= [ 1 11 21 -1 0 2 -3 . 1 2 4 8 2 |
Problem 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of...
Prove that mutual exclusion is satisfied by the following
algorithm for the critical section problem:
(image attached)
Note that the if statement is atomic.
By "if statement is atomic" it means that lines p2 and q2 are
either executed in their entirety once execution begins. No
interleaving is possible for example, between the "if" test and the
assignment of wantp/wantq.
You must prove the appropriate program invariant with the help of
induction. It's okay to prove lemmas through program
invariant-induction...
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