DIS MATH CHP 5

Let P(n) be the statement that the sum of the first n positive odd integers is n2. Put the following steps in order for the proof by mathematical induction that P(n) is true for all positive integers n.

1. P1 2.assume 3now 4.by the 5. and by 6. Therefore

Put the following steps in order to make a correct template for proofs by mathematical induction.

1. express 2. base step 3. inductive step 4. state what needs 5.prove that 6.write this compl 7. state

Why is mathematical induction a valid proof technique? Put the following steps of a proof of such in the correct order.

1.supose 2.to show 3.then the 4.we know tha t 5.furthermore 6.since p(m

The rule of inference behind mathematical induction can be expressed in predicate logic as:

(P(1)Λ∀k (P(k)→P(k+1)))→∀n P(n)

What is the inductive hypothesis in a proof by mathematical induction that n3 - n is divisible by 3 whenever n is a positive integer?

We assume that k3 - k is divisible by 3 for an arbitrary positive integer k.

Which of these statements can be proved using mathematical induction? Multiple select question.

n < 2n for all positive integers n. 20 + ... + 2n ≤ 2n + 1 - 1 for all positive integers n. A set with n elements has 2n subsets for all nonnegative integers n. 57 | 7n + 2 + 82n + 1 for all nonnegative integers n.

Which of these steps are part of a template for proofs by mathematical induction

Show that if P(k) is true for an arbitrary fixed integer k ≥ b, then P(k + 1) is true. Write "By mathematical induction, P(n) is true for all integers n with n ≥ b." Express the statement to be proved in the form "for all n ≥ b, P(n)" for a fixed integer b. Show that P(b) is true.

The inductive hypothesis is the statement that P(k) is true for the positive integer k. To complete the inductive step, we must show that if P(k) is true, then P(k + 1) must also be true. So, consider a set of k+ 1 distinct lines in the plane. By the inductive hypothesis, the first k of these lines meet in a common point p1. Moreover, by the inductive hypothesis, the last k of these lines meet in a common point p2. We will show that p1 and p2 must be the same point. If p1 and p2 were different points, all lines containing both of them must be the same line because two points determine a line. This contradicts our assumption that all these lines are distinct. Thus, p1 and p2 are the same point. We conclude that the point p1 = p2 lies on all k + 1 lines. We have shown that P(k + 1) is true assuming that P(k) is true. This completes the inductive step.

The proof is incorrect because the inductive step actually requires that k ≥ 3, since P(2) does not actually imply P(3).