Which of the following are considered members of a class
Associative
(P v Q) v R = P v (Q v R) (P ^ Q) ^ R = P ^ (Q ^/ R)
Empty Set
0
Proof by contradiction
1. Assume that P ^ Q' 2. Deduce a contradiction 3. Use when Q says something is not true.
Direct Proofs
1. Write out proof in english 2. (Hypothesis) Assume x,y are arbitrary integers as x,y are (odd or even) there exists some Ep : x= 2P+1 and Eq= 2Q+1 3. Then do math and prove 4. Do not need to show actual numbers just show that it is true.
Sum of degrees of vertices
2m m<= C(n,2)
Union
A set with all elements of both sets.
Subset
A single element in a set denoted by C_ includes brackets.
Complement '
A' set of objects that are not in A
Difference
A-B is the number of elements in A that are not in B.
Kruskals
Already build just connect edges
Antisymmetry
Any relation R on a set A is said to be antisymmetric if (a,b) R and (b,a) R, then a = b
Prims
Build the tree based on the least value
Pascal's Identity
C(n, k) = C(n-1, k) + C(n-1, k-1) where n>k >= 1
Element
Denoted by e means whole peice in set. DOES NOT INCLUDE { }.
Intersection
Denoted by n is the set of objects in A and B (A n B)
Cardinality
Denoted by | | is the number of elements in a set.
Weight of Spanning Tree
Equal to the sum of weights of edges in it.
Cartesian product
Every element of A times every element of B to make a new set for example A= {1,2,3} B={1,2} A x B= {1,1},{1,2}, {2,1}, {2,2} {3,1}. {3,2}
Symetry
Every such relation between two terms go back and forth. So that (a,b) and (b,a) are included in set S.
Path
Every two consecutive vertices are adjacent
Subgraph
G' = (V', E') is a subgraph of G = (V, E) if V' Í V and E' Í E.
Reflexive property
Goes back to itself in a loop.
Minimum Spanning Tree
Grow a minimum spanning tree T by starting with one vertex in T and adding a new vertex to T in each iteration. As long as T does not have all the vertices of the graph •Pick an edge x—y of least cost such that x is in T and y is not in T. •Add vertex y and edge x—y to T.
Cycle
Neither vertices or edged allowed to repeat
H.S
P -> Q ->R = p -> r
Implication
P -> Q = p' V Q
Communitive
P V Q = Q V p P^ Q= Q^P
M.P
P->Q if p is true then Q is implied to be true as well
Absorbtion
P^(p V Q) = P
Contraposition
Reverse order and opposite example: A^B -> C would be C'-> A' v B'
Converse
Reverse order of original statement ex: A^B -> C converse would be C -> A^B
Spanning tree
T = (VT, ET) is a spanning tree of a connected graph G = (V, E) if 1. T is a tree 2. T is a subgraph of G 3. VT = V Should hav N-1 edges at end Where N= number of vertices
Transitive
if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.