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Properties of Relations and an Euler Walk

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1. Determine which of the reflexive, symmetric, and transitive properties are satisfied by the given relation R defined over set S. See Appendix A for the definition of reflexive, symmetric, and transitive properties.
S={1,2,3} and R={(1,1), (1,2), (2,1), (2,2)}
Appendix A Definition
A relation R on a set S may have any of the following special properties.
(1) If for each x in S, x R x is true, then R is called reflexive.
(2) If y R x is true whenever x R y is true, then R is called symmetric.
(3) If x R z is true whenever x R y and y R z are both true, then R is called transitive.

(2) The city of Konigsberg, located on the banks of the Pregel River, had seven bridges that connected islands in the river to the shores as illustrated below. It was the custom of the town people to stroll on Sunday afternoons and, in particular, to cross over the bridges. The people of Konigsberg wanted to know if it was possible to stroll in such a way that it was possible to go over each bridge exactly once and return to the starting point. Is it?

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Properties of relations and an Euler walk are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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