(4). (25 points) The following problem contains a proposition and a sketch of an (weak) induction...
1. What is wrong with the following proof that shows all integers are equal? (Please explain which step in this proof is incorrect and why is it so.) Let P(n) be the proposition that all the numbers in any set of size n are equal. 1) Base case: P(1) is clearly true. 2) Now assume that P(n) is true. That is for any set of size n all the numbers are the same. Consider any set of n + 1...
Please answer with the details. Thanks!
In this problem using induction you prove that every finitely generated vector space has a basis. In fact, every vector space has a basis, but the proof of that is beyond the scope of this course Before trying this question, make sure you read the induction notes on Quercus. Let V be a non-zero initely generated vector space (1) Let u, Vi, . . . , v,e V. Prove tfe Span何, . . ....
DISCRETE MATHEMATICS
Problem 3 (10 points) Use mathematical induction to prove the following statement for all n 21. For full credit, mention the base case (1pt), the induction hypothesis (1 pt) and the induction step (8 pts). 12 22 32
QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE
r=1(no induction required, just use the definition of the
determinants)
Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
1. Consider the following sequence: a,-1+30-4 a,-1+30 +3a,-16 а,-1+3a0 + 3a1 +3a2-64 Use Weak Mathematical Induction (on homework 7A, you used Strong) to prove that an-4 for all n2 0. (a) State and prove the Base Case: (b) State the Inductive Hypothesis (c) Show the Inductive Step
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
3. (12 points) Consider the following sum: n Sn = {(i + 1)(i +2) i=0 (a) Use properties of summations to find a closed form expression for Sn. Simplify your answer into a polynomial with rational coefficients. Show your work, and clearly indicate your final answer. (b) Use weak induction to prove that your closed form works for every integer n > 0. Make sure you include all three parts, and label them appropriately!
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...
my question contains 4 points kindly answer the all, if u won't
don't solve , answer clear and if u r not sure from answer don't
solve plz , thank u , sir
An analyst wants to test if the mean peak loads (in MPa) of two alloys are different. He takes a random sample of n peak load measurements from alloy 1 and finds a sample average #7 =9.5 MPa, and a sample standard deviation s =0.82 MPa. He...
please answer all the questions.
just rearranging. Explanation is not needed.
Use modular arithmetic to prove that 3|(221 – 1) for an integer n > 0. Hence, 3|(221 – 1) for n > 0. To show that 3|(221 – 1), we can show that (221 – 1) = 0 (mod 3). We have: (221 – 1) = (4” – 1) (mod 3) Then, (22n – 1) = (1 - 1) = 0 (mod 3) Since 4 = 1 (mod 3),...