Explanation Approach: A subset A of the real numbers is said to be inductive if it contains the number 1, and if for every x in A , the number x + 1 is also in A . Let Ã be the collection of all inductive subsets of ℝ . Then the set ℤ + of positive integers is defined by the equation ℤ + = ∩ A ∈ Ã A . Principle of Mathematical Induction: The logic of the method is that a. if P n is true for n = 1 , and b. if the truth of P k always implies that P k + 1 is true, then the statement P n is true for all positive integers n . Proof: The proof of ‘For given n ∈ ℤ + , every nonempty subset of { 1 , ... , n } has a largest element’ by induction is as follows, Let A is the set of all positive integers n such that the above statement is true. For n = 1 , the set is { 1 , ... , n } = { 1 } . Since set { 1 } has only one nonempty subset, namely { 1 } itself that has largest element 1. So, every nonempty subset of the given set has a largest element. The given statement is true for n = 1

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ISBN: 9780134689517

Author: Munkres, James R.

Publisher: Pearson,

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Chapter 1.4, Problem 4.1E

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Every nonempty subset of {1,...,n} has a largest element by induction where n∈ℤ+.

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Chapter 1.4, Problem 3E

Chapter 1.4, Problem 4.2E

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To prove: Every nonempty subset of { 1 , ... , n } has a largest element by induction where n ∈ ℤ + .

Solution Summary: The author explains how to prove that every nonempty subset of left1,mathrm...,nright has a largest element by induction.

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To prove:

Every nonempty subset of {1,...,n} has a largest element by induction where n∈ℤ+.

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Chapter 1 Solutions

To prove: Every nonempty subset of { 1 , ... , n } has a largest element by induction where n ∈ ℤ + .

Solution Summary: The author explains how to prove that every nonempty subset of left1,mathrm...,nright has a largest element by induction.

Videos

To prove: Every nonempty subset of {1,...,n} has a largest element by induction where n∈ℤ+.

Want to see the full answer?

Chapter 1 Solutions

To prove:

Every nonempty subset of {1,...,n} has a largest element by induction where n∈ℤ+.