Homework Answers
(6) Let A denote an m x n matrix. Prove that rank A < 1 if...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S):...
In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S): the power set of a set S; namely, the set of all subsets of S. P*(S): = P(S) - {@}; namely, the set of all non-empty subsets of S. The following question is a challenging one! As a start, may be you try this question for small values of n, say n=1,2,3. Can you make a guess? (1) We all know that P*(On) has...
Question 1: Let the functions M(n) and f(n) be defined as follows. if n = 0...
Question 1: Let the functions M(n) and f(n) be defined as follows. if n = 0 (1, M(n) = {3}: M(n − 1) – 2n +1, if n > 0 f(n) = n +1 Prove that M(n) = f(n) for all n > 0.
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use...
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use induction to prove that F. ends in digit 7 whenever n 2 is an integer (b) Use induction to prove that F= 2 + IT- Fholds for all neN. (c) Use (b) to prove that ged(F, F.) = 1 holds for all distinct nonnegative integers m, na (d) Use (e) to give a quick proof that there must be infinitely many primes! That is...
Therom 1.8.2 n choose k = (n choose n-k) n choose k = (n-1 choose K)...
Therom 1.8.2
n choose k = (n choose n-k)
n choose k = (n-1 choose K) + (n-1 choose K-1)
2n = summation of (n choose i )
please use the induction method
(a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
7. Consider the following proposed sorting algorithm supersort (int n, int start, int end, keytype SI1)1...
7. Consider the following proposed sorting algorithm supersort (int n, int start, int end, keytype SI1)1 if(n > 1) { if (SIstart] > S[end]) swap SIstart] with Stend]; supersort(n-l, start, end-1, s) supersort (n-1, start+, end, S) a) 3 pts) Give a recurrence relation (with initial condition) that describes the complexity of this sort algorithm b) (4 pts) Solve the recurrence froma) c) (3 pts) Is supersort guaranteed to correctly sort the list of items? Justify your answer. (A formal...
Roots (20 points). Consider the loop-gain transfer function L(S) = TS-a)n-m where n and m are...
Roots (20 points). Consider the loop-gain transfer function L(S) = TS-a)n-m where n and m are integers such that n > m and a € R. Also, consider the characteristic equation 1+ KL(S) = 0, with 0 <KER, which can be equivalently written as nam (s– an-m + K = TI (s – rj) = 0. Show that num ri=(n - m), for any 0 <KER.
1 For each of the following pairs of numbers a and b, calculate and find integers...
1 For each of the following pairs of numbers a and b, calculate and find integers r and s such ged (a; b) by Eucledian algorithm that gcd(a; b) = ra + sb. ia= 203, b-91 ii a = 21, b=8 2 Prove that for n 2 1,2+2+2+2* +...+2 -2n+1 -2 3 Prove that Vn 2 1,8" -3 is divisible by 5. 4 Prove that + n(n+1) = nnīYn E N where N is the set of all positive integers....
0. For n E N, n > 1, let s, be defined by 8. *Let s1...
0. For n E N, n > 1, let s, be defined by 8. *Let s1 1 +S2m 2 S2m-1 S2m+1 $2m 2 Find lim s, and lim s,.
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...
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S): the power set of a set S; namely, the set of all subsets of S. P*(S): = P(S) - {@}; namely, the set of all non-empty subsets of S. The following question is a challenging one! As a start, may be you try this question for small values of n, say n=1,2,3. Can you make a guess? (1) We all know that P*(On) has...
Question 1: Let the functions M(n) and f(n) be defined as follows. if n = 0 (1, M(n) = {3}: M(n − 1) – 2n +1, if n > 0 f(n) = n +1 Prove that M(n) = f(n) for all n > 0.
For an integer n > 0, consider the positive integer F. = 22 +1. (a) Use induction to prove that F. ends in digit 7 whenever n 2 is an integer (b) Use induction to prove that F= 2 + IT- Fholds for all neN. (c) Use (b) to prove that ged(F, F.) = 1 holds for all distinct nonnegative integers m, na (d) Use (e) to give a quick proof that there must be infinitely many primes! That is...
Therom 1.8.2
n choose k = (n choose n-k)
n choose k = (n-1 choose K) + (n-1 choose K-1)
2n = summation of (n choose i )
please use the induction method
(a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
7. Consider the following proposed sorting algorithm supersort (int n, int start, int end, keytype SI1)1 if(n > 1) { if (SIstart] > S[end]) swap SIstart] with Stend]; supersort(n-l, start, end-1, s) supersort (n-1, start+, end, S) a) 3 pts) Give a recurrence relation (with initial condition) that describes the complexity of this sort algorithm b) (4 pts) Solve the recurrence froma) c) (3 pts) Is supersort guaranteed to correctly sort the list of items? Justify your answer. (A formal...
Roots (20 points). Consider the loop-gain transfer function L(S) = TS-a)n-m where n and m are integers such that n > m and a € R. Also, consider the characteristic equation 1+ KL(S) = 0, with 0 <KER, which can be equivalently written as nam (s– an-m + K = TI (s – rj) = 0. Show that num ri=(n - m), for any 0 <KER.
1 For each of the following pairs of numbers a and b, calculate and find integers r and s such ged (a; b) by Eucledian algorithm that gcd(a; b) = ra + sb. ia= 203, b-91 ii a = 21, b=8 2 Prove that for n 2 1,2+2+2+2* +...+2 -2n+1 -2 3 Prove that Vn 2 1,8" -3 is divisible by 5. 4 Prove that + n(n+1) = nnīYn E N where N is the set of all positive integers....
0. For n E N, n > 1, let s, be defined by 8. *Let s1 1 +S2m 2 S2m-1 S2m+1 $2m 2 Find lim s, and lim s,.
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...
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