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Chapter 1: Problem 12

Prove the Principle of Mathematical Induction \(1.2 .3\) (second version).

Short Answer

Expert verified
Proof has been done using the Second Principle of Mathematical Induction. Demonstrating the establishment base case, the satisfaction of the inductive step and reaching an inductive conclusion provides the proof of the principle throughout all positive integers.

Step by step solution

01

Base case

The first step in this process is to establish a 'base case', this is generally the smallest possible valid value of \(n\), quite often \(n = 1\). In Mathematical Induction (second version), the base case is verified.
02

Inductive step

The inductive step, assumes that the statement holds for some integer \(k\), i.e. \(P(k)\) is true, where \(P(k)\) stands for the statement we are trying to prove. Then, it has to be proved that the statement holds for \(k+1\), i.e. \(P(k+1)\) is also true. The key here is to think about how one could use the assumption that \(P(k)\) is true to show that \(P(k+1)\) is also true. This is done by substituting \(k+1\) anywhere \(k\) appears in the predicate \(P\). It may be necessary to rearrange or re-write some parts of the predicate as part of this process.
03

Inductive Conclusion

Lastly, it needs to be shown how these two steps together, demonstrate the inductive conclusion, that the principle holds for all positive integers. In a formal proof, the conclusion would mention some combination of the proposition, base case, and inductive hypothesis, and would explain how these demonstrate the whole proposition. If steps 1 and 2 hold, then by the second version of the principle of mathematical induction, the proposition is true for all positive integers.

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Most popular questions from this chapter

The symmetric difference of two sets \(A\) and \(B\) is the set \(D\) of all elements that belong to either \(A\) or \(B\) but not both. Represent \(D\) with a diagram. (a) Show that \(D=(A \backslash B) \cup(B \backslash A)\). (b) Show that \(D\) is also given by \(D=(A \cup B) \backslash(A \cap B)\).

Prove that \(5^{n}-4 n-1\) is divisible by 16 for all \(n \in \mathbb{N}\).

Let \(f(x):=1 / x^{2}, x \neq 0, x \in \mathbb{R}\) (a) Determine the direct image \(f(E)\) where \(E:=\\{x \in \mathbb{R}: 1 \leq x \leq 2\\}\). (b) Determine the inverse image \(f^{-1}(G)\) where \(G:=\\{x \in \mathbb{R}: 1 \leq x \leq 4\\}\).

Conjecture a formula for the sum of the first \(n\) odd natural numbers \(1+3+\cdots+(2 n-1)\), and prove your formula by using Mathematical Induction.

Use Mathematical Induction to prove that if the set \(S\) has \(n\) elements, then \(\mathcal{P}(S)\) has \(2^{n}\) elements.
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