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Consider the statement, For all connected graphs G, if every vertex of G has even degree, then G has an Euler circuit. Note, you do not need to know anything about graphs to answer these questions. What would the first line of a direct proof be? O A. Let G be a connected graph and assume it has an Euler circuit. O B. Let G be a connected graph and assume it does not have an Euler circuit. O C. Assume there is some connected graph that has all even degree vertices but does not contains an Euler circuit. O D. Let G be a connected graph and assume at least one vertex has odd degree. O E. Let G be a connected graph and assume every vertex has even degree. What would the first line of a proof by contrapositive be? O A. Let G be a connected graph and assume it has an Euler circuit. O B. Let G be a connected graph and assume every vertex has even degree. OC. Let G be a connected graph and assume it does not have an Euler circuit. O D. Let G be a connected graph and assume at least one vertex has odd degree. O E. Assume there is some connected graph that has all even degree vertices but does not contains an Euler circuit. What would the first line of a proof by contradiction be? O A. Let G be a connected graph and assume at least one vertex has odd degree. O B. Let G be a connected graph and assume it does not have an Euler circuit. O C. Let G be a connected graph and assume it has an Euler circuit. OD. Assume there is some connected graph that has all even degree vertices but does not contains an Euler circuit. OE. Let G be a connected graph and assume every vertex has even degree.

Consider the statement, For all connected graphs G, if every vertex of G has even degree, then G has an Euler circuit. Note, you do not need to know anything about graphs to answer these questions. What would the first line of a direct proof be? O A. Let G be a connected graph and assume it has an Euler circuit. O B. Let G be a connected graph and assume it does not have an Euler circuit. O C. Assume there is some connected graph that has all even degree vertices but does not contains an Euler circuit. O D. Let G be a connected graph and assume at least one vertex has odd degree. O E. Let G be a connected graph and assume every vertex has even degree. What would the first line of a proof by contrapositive be? O A. Let G be a connected graph and assume it has an Euler circuit. O B. Let G be a connected graph and assume every vertex has even degree. OC. Let G be a connected graph and assume it does not have an Euler circuit. O D. Let G be a connected graph and assume at least one vertex has odd degree. O E. Assume there is some connected graph that has all even degree vertices but does not contains an Euler circuit. What would the first line of a proof by contradiction be? O A. Let G be a connected graph and assume at least one vertex has odd degree. O B. Let G be a connected graph and assume it does not have an Euler circuit. O C. Let G be a connected graph and assume it has an Euler circuit. OD. Assume there is some connected graph that has all even degree vertices but does not contains an Euler circuit. OE. Let G be a connected graph and assume every vertex has even degree.

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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Consider the statement: "For all connected graphs \( G \), if every vertex of \( G \) has even degree, then \( G \) has an Euler circuit." Note, you do not need to know anything about graphs to answer these questions. --- **What would the first line of a direct proof be?** - \( \circ \) A. Let \( G \) be a connected graph and assume it has an Euler circuit. - \( \circ \) B. Let \( G \) be a connected graph and assume it does not have an Euler circuit. - \( \circ \) C. Assume there is some connected graph that has all even degree vertices but does not contain an Euler circuit. - \( \circ \) D. Let \( G \) be a connected graph and assume at least one vertex has odd degree. - \( \circ \) E. Let \( G \) be a connected graph and assume every vertex has even degree. --- **What would the first line of a proof by contrapositive be?** - \( \circ \) A. Let \( G \) be a connected graph and assume it has an Euler circuit. - \( \circ \) B. Let \( G \) be a connected graph and assume every vertex has even degree. - \( \circ \) C. Let \( G \) be a connected graph and assume it does not have an Euler circuit. - \( \circ \) D. Let \( G \) be a connected graph and assume at least one vertex has odd degree. - \( \circ \) E. Assume there is some connected graph that has all even degree vertices but does not contain an Euler circuit. --- **What would the first line of a proof by contradiction be?** - \( \circ \) A. Let \( G \) be a connected graph and assume at least one vertex has odd degree. - \( \circ \) B. Let \( G \) be a connected graph and assume it does not have an Euler circuit. - \( \circ \) C. Let \( G \) be a connected graph and assume it has an Euler circuit. - \( \circ \) D. Assume there is some connected graph that has all even degree vertices but does not contain an Euler circuit. - \( \circ \) E. Let \( G \) be a connected graph and assume every vertex has even degree.
Transcribed Image Text:Consider the statement: "For all connected graphs \( G \), if every vertex of \( G \) has even degree, then \( G \) has an Euler circuit." Note, you do not need to know anything about graphs to answer these questions. --- **What would the first line of a direct proof be?** - \( \circ \) A. Let \( G \) be a connected graph and assume it has an Euler circuit. - \( \circ \) B. Let \( G \) be a connected graph and assume it does not have an Euler circuit. - \( \circ \) C. Assume there is some connected graph that has all even degree vertices but does not contain an Euler circuit. - \( \circ \) D. Let \( G \) be a connected graph and assume at least one vertex has odd degree. - \( \circ \) E. Let \( G \) be a connected graph and assume every vertex has even degree. --- **What would the first line of a proof by contrapositive be?** - \( \circ \) A. Let \( G \) be a connected graph and assume it has an Euler circuit. - \( \circ \) B. Let \( G \) be a connected graph and assume every vertex has even degree. - \( \circ \) C. Let \( G \) be a connected graph and assume it does not have an Euler circuit. - \( \circ \) D. Let \( G \) be a connected graph and assume at least one vertex has odd degree. - \( \circ \) E. Assume there is some connected graph that has all even degree vertices but does not contain an Euler circuit. --- **What would the first line of a proof by contradiction be?** - \( \circ \) A. Let \( G \) be a connected graph and assume at least one vertex has odd degree. - \( \circ \) B. Let \( G \) be a connected graph and assume it does not have an Euler circuit. - \( \circ \) C. Let \( G \) be a connected graph and assume it has an Euler circuit. - \( \circ \) D. Assume there is some connected graph that has all even degree vertices but does not contain an Euler circuit. - \( \circ \) E. Let \( G \) be a connected graph and assume every vertex has even degree.
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