MATH 272
~(p -> q) == (p ^ ~q) • ~(If Chris gets a flu shot, then he will not get the flu) == Christ gets a flu shot, but he gets the flu.
How do you negate an implication? Give a written example
"Let" "Suppose"
How should you start a proof How do you NOT?
...A tree Building a communication network with the fewest cost
what is a weighted graph? what is prim's algorithm? and describe how to use it.
the number of edges connected to a given vertex
what is the definition of a "degree" in terms of graph theory
p -> q
what is the format for an implication?
closed trail begins and ends on same vertex repeat vertices, but no repeat edges (can repeat vertices)
what's a circuit?
only one vertex repeated (kinda like a fancy loop)
what's a cycle?
no repeated vertices
what's a path?
no repeating edges
what's a trail?
one vertex, no edges
what's a trivial circuit?
• ~(p v q) == ~p ^ ~q • ~(p ^ q) == ~p v ~q
Basic DeMorgan's?
Two vertices sharing one edge
Define Adjacent Vertices
G = V & E
Define a graph
Spot 1
In a Josephus Circle, if a game starts with 2^n people, and every other prson is eliminated, starting with P2, who will survive?
!!
Prove: The number F_3k is even for all positive integers k.
2.4
Show that nΣ(i=1) (4i -1) = n(2n+1) for every positive integer n
false loop a --- b loop
True or false, If true, prove, if false, provide a counter example In any connected graph G, if every vertex has degree 3, then g must contain a cycle of length 3
._____. a,1,b,1,a is counter
True or false, If true, prove, if false, provide a counter example In any graph G, if W is a closed walk, then W contains a cycle
false a -- b loop
True or false, If true, prove, if false, provide a counter example In any graph G, some pair of vertices must have the same degree
largest pieces of a graph
What are components? Give an example for a connected and non connected graph
a/b, b != 0
What are rational numbers?
"any integer ... is even or odd" "n is not divisible by m"
What are the key words of proving by cases?
i) P(1) is true ii) if P(k) is assumed to be true, then P(k+1) is true, for any positive integer k Then P(n) is true for every positive integer n
What are the steps of induction?
implication contrapositive
What are the two guaranteed ways to do a standard proof?
uses all edges on graph
What does it mean to be Eulerian?
all V and E in H are also in G
What does it mean to be a sub-graph?
in an graph, the number of vertices with odd degree is even
What is Corollary 4 of Graph Theory
~(pvq) == ~p ^ ~q ~(p^q) == ~p V ~q
What is DeMorgan's Law?
sum of degrees is two time the amount of edges in a simple graph, if there are n vertices on a graph, the SMALLEST amount of edges the graph can have is n-1
What is Theorem 3 of Graph Theory
A graph has an EULERIAN TRAIL if and only if it contains exactly two odd degree'd vertices MOREOVER, the trail must begin and end at these two vertices
What is Theory 5 of Graph Theory?
p <-> q p if and only if q only "true" when both T and F agree( are the same) p -> q = T q -> p = T Statement is true T F = F F T =F
What is a biconditional?
There is a walk between any two chosen vertices
What is a connected GRAPH?
Truth Table with all False values between two comaprisons
What is a contradiction?
~q -> ~p
What is a contrapositive?
q -> p
What is a converse?
lists the degrees from largest to smallest if there are n vertices on a graph, the the largest degree can only be n-1 in a simple graph, there cannot be an odd number of odd degree'd vertices two max vertices both sharing edges with the remaining vertices, where each vertex must be connected by two edges, a vertex of degree one is impossible
What is a degree sequence? What are the ways to determine if the degree sequence produces a graph?
A SIMPLE, CONNECTED GRAPH WITH NO CYCLES on a tree, vertices of degree 1 are called LEAVES
What is a graph tree?
Truth Table with all True values between two comparisons
What is a tautology?
p -> q
What is an implication?
~p -> ~q
What is an inverse
m*n +1 n boxes some box contains m+1 objects
What is the Pigeonhole Principle equation
an edge with one vertex only, that loops back on itself deg 2
What is the definition of a loop? how many degrees does it count as?
r = a/b, a and b are integers and b=/= 0
What is the definition of a real number r?
"a mod b = r" == "a = b * q + r", where "0 <= r < b"
What is the division theorem? What is its relation to modules?
P(x)
What is the notation for a general predicate?
2.3
[B] Let P(n) be the statement "nΣ(i=1) 2^(i-1) = 2^(n) -1 use induction to prove it's possible for every integer n
4
[B] Let n be any integer, calculate (25q^2 + 30q + 9) mod 5
2.4
[B] Show that n³+ 2n is divisible by 3 for all positive integers.
2.3
[B] a_K = a_(K-1) + (2K) for (K >= 2) and a_1 = 2 Show that a_n = n(n+1), for all (n >= 1)
2.4
[B] a_n = 2a_n-1 + a_n-2, where a_1 = 5 and a_2 =10. Use induction to show that for all (n>=3), a_n < 3^n.
2.3
[B] a_K = a_(K-1) + (2K -1) for (K >= 2) and a_1 = 1 Show that a_n = n^2, for all (n>=1)
v1e1v2e2 alternating vertex and edge closed = begins and ends with same vertex length = how many edges
define a walk what does it mean when it's closed what does a walk's length mean?
two edges, share same vertex
define parallel edges
notes
prove that every tree has at least two leaves
notes
prove that for n vertices in a tree T , T has "n-1" edges
T,F,F,F T,T,T,F
the difference between an "and" and an "or" truth table?
sigma 1+2+....n == n(n+1)/2
what are the forms for the closed formula for sums?
graph with no loops or prarallel edges [] ()
what is a simple graph? what is the notation for unordered V and E directed/ordered?
...a tree (subgraph) that uses every node in the graph
what is a spanning tree?
