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.