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. 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.

Advanced Engineering Mathematics
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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**Inductive Proof of Trees with n Vertices Having n-1 Edges** **Statement**: For all positive integers \( n \), every tree with exactly \( n \) vertices has exactly \( n - 1 \) edges. **Proof by Induction**: Let \( P(n) \) be the statement 'every tree with exactly \( n \) vertices has exactly \( n - 1 \) edges'. (a) **Prove the base case.** (b) Assume \( P(k) \) is true for some \( k \geq 1 \); 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. **Claim**: \( G \) has a leaf. *Explain why.* ii. Let \( v \) be a leaf in \( G \). Form a new graph \( G' \) by deleting \( v \) 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. This inductive approach establishes that any tree with a certain number of vertices will always exhibit this specific relationship between vertices and edges.
Transcribed Image Text:**Inductive Proof of Trees with n Vertices Having n-1 Edges** **Statement**: For all positive integers \( n \), every tree with exactly \( n \) vertices has exactly \( n - 1 \) edges. **Proof by Induction**: Let \( P(n) \) be the statement 'every tree with exactly \( n \) vertices has exactly \( n - 1 \) edges'. (a) **Prove the base case.** (b) Assume \( P(k) \) is true for some \( k \geq 1 \); 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. **Claim**: \( G \) has a leaf. *Explain why.* ii. Let \( v \) be a leaf in \( G \). Form a new graph \( G' \) by deleting \( v \) 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. This inductive approach establishes that any tree with a certain number of vertices will always exhibit this specific relationship between vertices and edges.
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