. In answering the questions below, you will prove the statement Statement. For all positive integers n, every tree with n exactly n vertices has exactly n - 1 edges. We will prove this by induction. So let P(n) be the statement 'every tree with exactly n vertices has exactly n - 1 edges'. (a) Prove the base case. (b) In the following, assume P(k) is true for some k ≥ 1; that is, every tree with exactly k vertices has exactly k - 1 edges. We will show that P(k+ 1) is true. So assume G is a tree with exactly k+1 vertices (so G has at least two vertices). We will show that G has exactly k edges. i. I claim G has a leaf. Explain why. ii. Let v be a leaf in G. Form a new graph G' by deleing G v from G. Is G' a tree? Why or why not? (use the definition to answer this). iii. Use G' to prove that G has exactly k edges.

. In answering the questions below, you will prove the statement Statement. For all positive integers n, every tree with n exactly n vertices has exactly n - 1 edges. We will prove this by induction. So let P(n) be the statement 'every tree with exactly n vertices has exactly n - 1 edges'. (a) Prove the base case. (b) In the following, assume P(k) is true for some k ≥ 1; that is, every tree with exactly k vertices has exactly k - 1 edges. We will show that P(k+ 1) is true. So assume G is a tree with exactly k+1 vertices (so G has at least two vertices). We will show that G has exactly k edges. i. I claim G has a leaf. Explain why. ii. Let v be a leaf in G. Form a new graph G' by deleing G v from G. Is G' a tree? Why or why not? (use the definition to answer this). iii. Use G' to prove that G has exactly k edges.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 33EQ

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