WGU C959

Elementary Row Operations

1. (Replacement) Replace one row by the sum of itself and a multiple of another row. 2. (Interchange) Interchange two rows 3. (Scaling) Multiply all entries in a row by a nonzero constant

Echelon Form

1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros.

Reduced Echelon Form

1. The leading entry in each nonzero row is 1 2. Each leading 1 is the only nonzero entry in its column

Cardinality

2^n, n = the number of elements in a set

Conjunctive Normal Form (CNF)

A Boolean expression that is a product of sums of literals.

Disjunctive Normal Form (DNF)

A Boolean expression that is the sum of products of literals.

Cycle

A circuit with the added restriction that no vertices are repeated (besides the first and last)

Partition

A collection of nonempty sets whose union is all of A and each pair in the collection is pairwise disjoint. Hint: Think of a partition as chopping the set up into disjoint pieces.

Weighted Graph

A graph in which there is a number associated with each edge [its weight].

Minterm

A product term having exactly one literal for every function variable

Proof by cases

A proof broken into separate cases, where these cases cover all possibilities (used for cases that involve integer parity--odd/even)

Identify Matrix

A square matrix where all the elements of the principal diagonal are ones and all other elements are zeros.

Subgraph

A subgraph is a subset of a graph's edges (and associated vertices) that constitutes a graph.

Spanning Tree

A subgraph of a connected graph that is a tree and includes all the vertices of the original graph.

Subset

A subset of a set consists of elements from the given set.

Proper Subset

A subset that does not contain every element in another set.

Path

A trail without repeated vertices

Rooted tree

A tree in which one vertex has been designated as the root and every edge is directed away from the root.

Closed walk

A walk that begins and ends at the same vertex

Open Walk

A walk that starts and finishes at different vertices.

Circuit

A walk where the first and last vertices are the same; no edge occurs more than once

Breadth-Search

Add the edges {a,e}, {a,b}, and {a,f}. Find the neighbors of e. The only neighbors of e are a and b, which are already included, so go to the next vertex, b. Find the neighbors of b. Since the only neighbors of b are e, a, and f, which are already included, go to the next vertex, f. Find the neighbors of f. The neighbors of f are c and d, both of which have not been added. Add the edges {f,c} and {f,d}. Repeat this procedure on c and d. Since the neighbors of the vertices c and d have been added, you have completed the search.

Regular graph

All vertices have the same degree

Proof by Contradiction (Indirect Proof)

Assume the theorem is false and then show how, under this assumption, a mathematical contradiction will arise (use same steps as direct proof).

Cn Graph

Cn is called a cycle on n vertices. The edges connect the vertices in a ring.

Injective

Describes a mapping in which each element of the domain maps to a single element of the range. Also, one-to-one.

Surjective

Describes a mapping in which each element of the range is the target of some element of the domain. Also, onto.

Bijective

Describes a relation that is both injective and surjective (one-to-one and onto).

In-degree

Description of a vertex; it is the number of edges pointing into it

Out-degree

Description of a vertex; it is the number of edges pointing out of it

Adjacent

Edge between two vertices

Parallel edges

Edges that connect the same pair of vertices

Literal

Element

Proof by counterexample

Find one counterexample (example to disprove the statement) to the claim.

Formula to find Number of Terms in Arithmetic Sequence

Find the common difference Plug into the formula the last term and the common difference for d and then solve for n in an=a1+(n−1)d.

unique paths in trees

Given any two vertices in a tree, there is only one path between them

Transitive

If a=b and b=c, then a=c

Pairwise Disjoint

If every pair of distinct sets in a sequence of sets is disjoint.

Matrix (Scalar) Multiplication

If r is a number, called a scalar, and A is a matrix, then rA is the matrix obtained by multiplying every entry in A by the scalar r. The result is called a scalar multiple of A.

Hasse Diagram

In a Hasse diagram, each element is shown by a labeled point. However, unlike digraphs, Hasse diagrams have very few edges - only lines depicting relative precedence of the elements. To create a Hasse diagram, use the following rules: If x ⪯ y, then make the x element appear lower in the diagram than y. If x ⪯ y and there is no z where x ⪯ z or z ⪯ y, then draw a segment connecting x and y. In Hasse diagrams, self-loops are removed and the transitive and reflexive properties are assumed.

Undirected graph theory

In an undirected graph, the total degree of the graph G is 2 times the number of edges in G.

Complete graph (n-clique)

Kn is called the complete graph on n vertices. Kn has an edge between every pair of vertices. The figure shows K6. Kn is sometimes called a clique of size n or an n-clique.

K n,m Graph

Kn,m has n+m vertices. The vertices are divided into two sets: one with m vertices and one set with n vertices. There are no edges between vertices in the same set, but there is an edge between every vertex in one set and every vertex in the other set.

Edge

Line between two endpoints; instance

Identity Function

Maps a set and all elements to itself

Degree

Number of neighbors

Depth-Search

Pick one of the neighbors of a-for example, e-and add the edge {a,e}. Pick a neighbor of e that has not already been scanned-for example, b-and add edge {e,b}. Pick a neighbor of b that has not already been scanned, say f, and add the edge {b,f}. Pick a neighbor of f that has not already been scanned, say c, and add the edge {f,c}. Pick a neighbor of c that has not already been scanned, say d, and add the edge {c,d}.

Equivalence Relations

Relations that are reflexive, symmetric, and transitive

Descendant/child

Same as ancestor/parent, but in downward direction

Set Z

Set of all integers

Set R

Set of all real numbers

Power Set

Set of all subsets of a set

Set Q

Set of rational numbers

Proof by exhaustion

Shows claim is true by plugging each element and determine its truth value.

Direct Proof

Starts with a given statement (hypothesis) and argues to a conclusion

Gaussian elimination

Step 1: Find the current pivot of the first column. The current pivot is the first nonzero entry in a matrix column. Step 2: If the current pivot is not in the first row, switch the first row with the row that contains the current pivot. Step 3: Perform elementary row operations using the current pivot so that all entries below the current pivot are zeros. Step 4: Repeat steps 1-3 for the submatrix containing all entries below and to the right of the current pivot. Step 5: Stop when the entries below each pivot are zero.

Gaussian Back Substitution

Step 1: Identify the corresponding system to the augmented matrix in reduced echelon form. Step 2: Solve the last equation for the basic variable. If the system does not contain any free variables, then the basic variable has a unique value. If the system contains free variables, then write the basic variable in terms of the free variable. Step 3: Substitute the unique value or expression for the variable into the equation above to obtain the solution for a different variable. Step 4: Repeat steps 1-3 until a unique value or expression is found for all variables.

Gauss-Jordan Method

Step 1: Perform forward elimination. Step 2: Scale the pivots by multiplying each row by the reciprocal of the corresponding pivot entry. This step should result in having 1s along the diagonal. Step 3: Perform elementary row operations using the rightmost pivot so that all entries above the rightmost pivot are zeros. Step 4: Repeat step 3 for the sub matrix containing all entries above and to the left of the rightmost pivot. Step 5: Stop when the entries above all pivots are zero.

Total Degree

Sum of all degrees

Partial Order Set (Poset)

The domain of a partially ordered set (poset) is denoted by (A, ⪯).

Exponential Function

The exponential function expb:R → R+ is defined as: expb(x)=bx where b is a positive real number and b ≠ 1. The parameter b is called the base of the exponent in the expression bx. The input x to the function bx is called the exponent.

Invertible

The inverse of f is obtained by exchanging the first and second entries in each pair in f. The inverse of f is denoted by f−1. If a function with an inverse has a finite domain, you can find the inverse by swapping the order in all the ordered pairs. Inverses for functions on infinite domains can be solved algebraically.

Properties of logarithms

The log of a product is equal to the sum of the log of the first base and the log of the second base The log of a quotient is equal to the difference of the logs of the numerator and denominator The log of a power is equal to the power times the log of the base The log of a new base is the log of the new base divided by the log of the old base in the new base

Logarithmic Function

The logarithm function is the inverse of the exponential function. For real number b > 0 and b ≠ 1, logb:R+ → R is defined as: bx=y ⇔ logb⁡y=x The parameter b is called the base of the logarithm in the expression logb y.

Hypercube Graph

The n-dimensional hypercube, denoted Qn, has 2n vertices. Each vertex is labeled with an n-bit string. Two vertices are connected by an edge if their corresponding labels differ by only one bit.

nth partial sum of a geometric sequence

The nth partial sum of a sequence is the sum of a finite number of consecutive terms beginning with the first term.

Ancestor/Parent Vertex

The parent of a vertex v is the first vertex encountered on a path from v to the root. Every vertex in the path from v to the root is called an ancestor. The parent is the first encountered ancestor on the path.

Post-Order Traversal

The process of systematically visiting every node in a tree once, starting at the root and proceeding left down the tree, accessing the first node encountered at its "right" side, proceeding likewise along the tree, accessing each node as encountered at its "right" side.

Pre-Order Traversal

The process of systematically visiting every node in a tree once, starting with the root node, proceeding to the left along the tree and accessing the node when the "left" side of the node is encountered.

Tree Traversal

The process of systematically visiting every node in a tree once. The three most common traversals are: pre-order, in-order, and post-order.

Set N

The set of natural numbers: All integers greater than or equal to 0

Cartesian Product

The set of ordered pairs whose first coordinate comes from set A and the second from set B. It is denoted A×B = {(a,b)|a∈A, b∈B}.

Walk

There are no restrictions on vertices

Matrix Multiplication

To get the entry of the product of two matrices, denoted (AB)ij , multiply the first row of A with The first column of B. This is called the row-column rule. Matrices do not have to be the same size to be multiplied, but their inner dimensions must match.

Proof by contrapositive

To prove "If A, then B": Assume "if not B, then not A" and use a direct proof to prove the original statement is true.

Matrix addition and subtraction

Two matrices can be added (or subtracted) if they are the same size (same number of rows and columns) by adding or subtracting each corresponding entry.

Disjoint

Two sets are disjoint if their intersection is empty. (A ∩ B = ∅)

Geometric Sequence Sum Formula

Use the following steps to find the sum of a geometric series: Identify a1,r,n Substitute values for a1,r and n into the formula Sn=a(1−rn1−r) Simplify the results

Arithmetic Sequence Sum Formula

Use the following steps to find the sum of an arithmetic series: Identify a1,an Determine n Substitute values for a1,an and n into the formula Sn=n(a1+an)2 Simplify the results

Neighbor

Vertices connected by an edge

Leaf

Vertices with no children

Siblings

Vertices with same parent

Properties of exponents

When you multiply powers with the same base, add the exponents When you multiply powers with the same base, add the exponents When you divide powers with the same base, subtract the exponents When you raise a product to a power, raise each factor with a power

Partial Order Relation

a partial order relation if it is transitive, reflexive, and anti-symmetric.

Trail

a walk with no repeated edges

Reflexive

a=a

D-regular graph

all the vertices have degree d

Arithmetic Sequence/Formula

an=a1+(n−1)d. Add the common difference to the first term to find the second. Add the common difference to the second to find the third, and so on. Write the terms separated by commas within brackets.

Geometric Sequence/Formula

an=cr^n. Divide each term by the previous term Compare the quotients. If the quotients of every pair of consecutive terms is the same, then the sequence is geometric.

Anti-Reflexive

for every x ∈ A, it is not true that xRx

Anti-Symmetric

for every x,y ∈ A, xRy and yRx imply that x = y

Symmetric

if a=b, then b=a

Minimum Spanning Tree

is a spanning tree T of G whose weight is no larger than any other spanning tree of G.

Free tree

no particular organization of the vertices and edges

Subtree

rooted at vertex v is the tree consisting of v and all v's descendants

Height

the highest level of a vertex

Transitive Closure

the smallest relation that is both transitive and includes all the pairs from R To find the transitive closure of relation R, start with the pairs in R and add pairs until you cannot add any more pairs.

Level

the vertex distance away from the root