# Properties of Relations and an Euler Walk

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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#### Solution Summary

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.