Answered: Is "If for some integer k P(k) is true,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Write a clear formal proof using contradiction
Suppose that P(1), P(2),... are statements. Assume that you know only the following: • P(3) is true. • For any k > 1, P(k) ⇒ P(k+2) is true. Find all n for which you can guarantee that P(n) is true. Justify your answer.
Let K be an integer satisfying 1 ≤ K ≤ N. In how many ways can we arrange the N objects in B so that B[K] is smaller than all the objects that follow it in B (i.e., B[K] < B[I] for all subscripts I greater than K)? Answer the question for N = 6, K = 4 first. Then give the answer for general N and K. Simplify your answer algebraically as much as possible.
I'm supposed to evaluate the proof, edit if there are errors or confirm that it is correct
Decide whether or not the following propositions are true or false. Justify each ofyour conclusions with a proof or a counterexample.(a) ∃x ∈ R (x 2 = x 3 and x > 0)(b) ∀n ∈ N + (2 n + 3 is prime)(c) ¬∃x ∈ R (x 2 < 36)
Give a carefully worded proof for the following statements using direct proof, contrapositive proof or proof by contradiction. No matter which proof method you choose to use, make sure you follow the outline properly. (If-and-Only-If Proof) Given an integer a and b, then b2(a + 3) is even if and only if a is odd or b is even. (Use the basic definitions of even and odd in your proof.)
Suppose that P(n), Q(n) and R(n) are statements about the integer n. Let S(n) be the statement: “If P(n) is true then Q(n) and R(n) are both true." (a) The contrapositive of S(n) is the statement Olf Q (n) and R(n) are both true then P(n) is false. Olf Q (n) and R(n) are both true then P(n) is true. Olf at least one of Q(n) and R(n) is false then P(n) is false. Olf Q (n) and R(n) are both false then P(n) is false. Olf at least one of Q(n) and R(n) is false then P(n) is true. Olf P(n) is false then at least one of Q(n) and R(n) is false. Olf Q (n) and R(n) are both false then P(n) is true. ONone of the other answers. (b) The statement: "If Q (n) and R(n) are both true then P(n) is true." is OThe converse of S(n). OThe contrapositive of S(n). OThe converse of the contrapositive of S(n). ONone of the other answers.
Let P be some predicate. Check the box next to each scenario in which Vn N, P(n) must be true. P(0) holds and for every natural number k > 0, if P(k) does not hold, then there is some natural number i 0, if P(i) holds for every natural number i
Suppose that P(n) is a propositional function. Determine for which nonnegative integers n the statement P(n) must be true if P(0) and P(1) are true and, for all nonnegative integers n, if P(n) is true, then P(n+3) and P(n+ 5) are true.
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) =…
For each of the following existential statements, indicate if the statementis true or false. If it is true, use a specific integer and show why it makes the statementtrue.(a) (∃n ∈ Z) (|n+5| / |n| = 2(b) (∃n ∈ Z) ((n^2 ) + 3n <0) (c) (∃n ∈ Z) (4n is odd)
If the contradiction method is used to prove the assertion "For any positive integer n > 0, if a" is even, then a is even.", then a valid implementation of that method begins by assuming that the negation of the statement is true. Choose all of the following statements that correctly express the negation of the above assertion. Vn E N – {0}, rem(a?, 2) = 0 → rem(a, 2) = 0 Vn EN – {0}, rem(a², 2) = 0 rem(a, 2) = 1 En e N – {0}, rem(a?, 2) = 0 → rem(a, 2) In eN - {0}, rem(a², 2) = 0 ^ rem(a, 2) = 1 For any positive integer n > 0, if a" is even, then a is odd. There exists a positive integer n > 0 where a" is even implies that a is od. There exists a positive integer n > 0 where a" is even and a is odd.

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Is "If for some integer k P(k) is true, then P(k+1) is true" logically equivalent to "For all integers k, if P(k) is true then P(k+1) is true."