lect_33 - Relations Margaret M Fleck 19 April 2010 This...

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Unformatted text preview: Relations Margaret M. Fleck 19 April 2010 This lecture introduces relations and covers basic properties of relations, i.e. parts of section 8.1 and 8.3 of Rosen. When you look at Rosen, be aware that we’re covering only relations on a single set, where he also covers relations between two sets. 1 Announcements Third honors homework is due this Wednesday (21st). Quiz 3 is coming up next Wednesday (28th). 2 More Connectivity Before I forget, notice that the vertices in a graph are also often called “nodes.” Recall that, in an undirected graph, a path of length k from vertex a to vertex b is a sequence of edges that connect end-to-end ( v 1 ,v 2 ) , ( v 2 ,v 3 ) , ( v 3 ,v 4 ) ,..., ( v k- 1 ,v k ) , where v 1 = a , v k = b . 1 An undirected graph G is connected if there is a path between every pair of vertices in G . That is, for any vertices a and b in G , there is a path from a to b . Three special types of paths are important to know about • A circuit is a path that ends at the same vertex where it started. • A path is simple if no edge occurs more than once in the path. • An Euler circuit of a graph G is a simple circuit that contains every edge in the graph. Fascination with Euler circuits dates back to the 18th century. At that time, the city of K¨onigberg, in Prussia, had a set of bridges that looked roughly as follows: Folks in the town wondered whether it was possible to take a walk in which you crossed each bridge exactly once, coming back to the same place you started. This is the kind of thing that starts long debates late at night in pubs, or keeps people amused during boring church services. Leonard Euler was the one who explained clearly why this isn’t possible....
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lect_33 - Relations Margaret M Fleck 19 April 2010 This...

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