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MATH 215 Ball State Review3Exam

Mathematical Sciences
Course Code
MATH 215
Study Guide

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Maths 215 Exam II Review Sheet
READ THE BOOK. Re-read every section of the covered material.
Re-do all the homework problems (don’t just look over your old solutions).
Know the following terms and symbols and how they are used.
Undirected/Directed graph (via diagram and formal
Vertices and Edges
Loops and Multiple edge
Euler path/circuit (find them when they exist)
Hamilton path/circuit (find them when they exist)
Graph coloring
Planar graph
Four color Theorem
Network and weighted edges
Binary search tree
Root node
Simple graph
Simple path
Complete graph
Set theory notation
Set builder notation
Be able to translate between set theory notation and
logic notation
Symbols ,, /, , ,,,\(set difference)
Note that 5 Nis False, but 5 Nis True.
Universal set
Cartesian product use {} and () properly.
Ex. {0,1} × {a} ̸={{0, a},{1, a}},
but {0,1} × {a}={(0, a),(1, a)}
Power set
DeMorgans Laws for sets
Inclusion-exclusion principle
What is the structure of a proof to show AB?
Function, domain, codomain, well-defined, one-to-one
function, onto, injection, surjection, bijection, one-to-
one correspondence, image
What is the structure of a proof to show a map is
onto? one-to-one?
Inverse of a function
Restriction of a function
Composition of functions
Reflexive, symmetric, transitive, anti-symmetric
Represent a relation via a graph
Partition and equivalence relation (and their connec-
Equivalence classes
Modular arithmetic
Partial order and total order
Hasse Diagram and topological sorting
Maximal/minimal elements
Isomorphic graphs
Theorems related to degree counting, Euler circuits
Tree (depth of vertex and height of tree)
Know and be able to apply the following theorems
of Euler:
1. In any graph, the sum of the degrees of the vertices
equals twice the number of edges. (Theorem 2.6)
2. If a graph has more than two vertices of odd degree, it
does not have an Euler path. (Theorem 2.12, HW # 18
in 2.6)
3. If a connected undirected graph has exactly two
vertices, vand w, of odd degree, then there is an Euler
path from vto w. (Theorem 2.11, HW # 17 in 2.6, text
mistakenly omitted adjective undirected)
4. If all the vertices of a connected undirected graph
have even degree, then the graph has an Euler circuit.
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