Part I: InductionProve each of the following statements using induction, strong induction,or structural induction. For each statement, answer the following questions.а.Complete the basis step of the proof.b.What is the inductive hypothesis?C.What do you need to show in the inductive step of the proof?d.Complete the inductive step of the proof.1. Let S be the set of perfect binary trees, defined as:• Basic step: a single vertex with no edges is a perfect binary tree T,.• Recursive step: if T1 and T2 are perfect binary trees of the same height, then a newperfect binary tree T' can be constructed by taking T1 and T2, adding a new vertexv, and adding edges between v and the roots of T, and T2.Prove that h(T) = log2(n(T) + 1) – 1 for any perfect binary tree T, where n(T) is the numberof vertices of T and h(T) is the height of T.Hint: Remember that h(T,) = h(T2) and n(T,) = n(T,). Also remember that h(T') = h(T,) + 1 andn(T') = n(T,) + n(T2) + 1.

Given- Let S be the set of perfect binary trees, defined as: Basic step: a single vertex with no…

Answered: Part I: Induction Prove each of the…

Advanced Engineering Mathematics

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Let, a1 = 1, az = 2, b1 = 0, b2 = 1, %3D for n 2 3, 2ап-1 + bn-1 for n 22 b, = bn–1+ an-1 an express an in terms of n.
Prove each of the following statements using induction, strong induction,or structural induction. For each proof, answer the following questions:• Complete the basis step of the proof.• What is the inductive hypothesis?• What do you need to show in the inductive step of the proof?• Complete the inductive step of the proof.
1.) Mathematical induction is useful in determining whether the statement REMAINS TRUE with any natural number as k. Is the statement TRUE or FALSE? 2.) We have only one (1) step that we may utilize in Mathematical induction – the Inductive step. Is this statement TRUE or FALSE? a. Underline only the PART that makes the following statement erroneous: "The Basis step helps in determining whether the first iteration of the statement, which is P(k+10), is TRUE." 3.) In a statement in Mathematical induction, we may only use positive integers or natural numbers for k. Is this statement TRUE or FALSE? 4.) We use k and k+1 for the Inductive step and Basis step respectively. Is this statement TRUE or FALSE? 5.) Prove that given any integer for n, n³ + 2n will be divisible by 3. n²(n+1)? 6.) 13 + 23 + 33 + +n³ ... 4 1 7.) 1(2) 1 + + 2(3) 1 + 3(3) 1 + п(п+1) given that any integer for n is positive. ... n+1' 8.) Write the formal proof that 2"+2 + 32n+1 is divisible by 7 for all positive…
Use mathematical induction to prove the following statement for all positive integers n. Fill in the appropriate information as you complete this proof. (a")" = a' тп a) The base case is the statement: b) The inductive assumption (in terms of k) is that: c) To complete the inductive proof, what should we do to this inductive assumption? A. Add am to both sides B. Multiply both sides by am C. Multiply both sides by a" D. Add a" to both sides
Consider the statement: For any positive integer nn, n3+2nn3+2n is a multiple of 33. Set up the framework for a proof by mathematical induction by doing the following steps: State the base case, and prove that the base case is true. Clearly state the inductive hypothesis. Clearly state the inductive step (that is, clearly state what needs to be proven following the inductive hypothesis). Give at least one suggestion for how a full proof of the inductive step might go, that is reasonable and likely to be useful.
Exercise 5.52 asked for a proof of the following statement: For each integer n > 12, there exist nonnegative integers a and b such that n = 3a + 7b. Use the Strong Principle of Mathematical Induction to give an alternative proof of this statement.
This part will explore one of the underlying mathematical ideas for a proof by induction. Assume that TCN and assume that 1 € T and that T is an inductive set. Use the definition of an inductive set to answer each of the following: (a) Is 2 € T? Explain. (b) Is 3 = T? Explain. (c) Is 4 € T? Explain. (d) Is 100 € T? Explain. (e) Do you think that T plain. = N? Ex-
1. Suggest a formula for each expression, and prove your hypothesis using the principle of mathematical induction, for all positive integers n. a. 3+ 32 + 33 +.. + 3" b. 1. 2+3 • 4 + 5• 6+ .. + (2n- 1)(2n)
Consider the statement that 3 divides n³ + 2n whenever n is a positive integer. Outline the proof by clicking and dragging to complete each statement. Let P(n) be the proposition that Basis step: P(1) states that Inductive step: Assume that Show that We have completed the basis step and the inductive step. By mathematical induction, we know that vk>0, (P(K)→ P(k+ 1)) is true, that is, vk>0, (3 divides k³ + 2k→ 3 divides (k+ 1)³ + 2(k+ 1)). 3 divided K³ + 2k for an arbitrary integer k> 0. P(n) is true for all integers n ≥ 1. 3 divides 1³ +2 3 divides n³ + 2n. 1, which is true since 1³ +21=3, and 3 divides 3. Reset Click and drag statements to fill in the details of showing that VK(P(K) → P(k+ 1)) is true, thereby completing the induction step. By the inductive hypothesis, 3 divides (k³ + 5k). 3 divides 3(K³ + 1) because it is 3 times an integer. By part (i) of Theorem 1 in Section 4.1, 3 divides the sum (k³ + 2k) + 3(k² + k + 1). This completes the inductive step. (k+ 1)³ + 2(k+ 1) =…
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10) Identify the mistake in the following “proof" by mathematical induction. (Fake) Theorem: For any positive integer n, all the numbers in a set of n numbers are equal to each other. Proof (by mathematical induction): It is obviously true that all the numbers in a set consisting of just one number are equal to each other, so the basis step is true. For the induction step, let A = {a1, a2, a3, . .., an41} be any set of n +1 numbers. Form two subsets each of size n: B = {a1,a2, A3, . numbers in A except an+1, and C consists of all the numbers in A except a2.) By the induction hypothesis, all the numbers in B equal a1 and all the numbers in C equal a1, since both sets have only n numbers. But every number in A is in B or C, so all the numbers in A equal a1; thus all are equal to each other. .., a,} and C = {a1, az, a4, . .. , an+1}. (Precisely, B consists of all the

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