What is Discrete Mathematics? Is it Discrete or Discreet? What does Discrete Mean?

Continuo us Discrete

Can you take a walk so that you cross over each bridge exactly once and wind up back where you started?

It might help to label the land masses so you can represent paths by strings of letters….

In Mathematics we like to abstract away from the details and only show the essentials…. This is an example of a (multi) graph which consists of edges and vertices.

Vertices: A, B, C, D Edges: (AB) 1, (AB) 2, (AC) 1, (AC) 2, (AD), (BD), (C,D). Note: (DA) = (AD) this is not a DiGraph. (AB) 1 (BD)(DC) is a path. (AB) 1 (BD)(DC)(CA) 1 is a cycle. Is there an Euler Cycle? An Euler Path?

What about this town? What if you had time to build one bridge?

Euler Industries 1 is a new computer company that claims to have designed a revolutionary new computer memory. The main memory is broken down into 4 sub-memory cells α, β, δ, and γ. These sub-memories can pass information back and forth along Euler Industries’ new cyberfluxors. There are two cyberfluxors connecting α and β and two cyberfluxors connecting β and δ. The three remaining cyberfluxors connect γ to α, β, and δ, respectively. That is, there is a cyberfluxor between γ and α, one between γ and β and between γ and δ. Each cyberfluxor is capable of carrying information in both directions. Euler Industries’ design depends on one important fact. A given piece of information must be able to pass through each cyberfluxor exactly once. Furthermore, the information must be able to return to the starting cell. Can the computer work or is there a flaw in the design? Compare this problem to the Königsburg Bridge problem. What information is necessary in order to solve each problem and what is extraneous? 1 This problem is reproduced with permission from an exercise written by Michael J. Bardzell.