C959 Flashcards (Master set)

Q

The set of rational numbers: All real numbers that can be expressed as a/b, where a and b are integers and b ≠ 0. For example, 0, 1/2, 5.23, -5/3

R

The set of real numbers. Real numbers can be a continuous quantity. For example, 0, 1/2, 5.23, -5/3, π, √2

Total degree of Graph

The sum of the degrees of all of the vertices. For example, the total degree of the picture is 2 + 3 + 3 + 2 + 2 = 12. (Hint: Just multiply the number of edges by 2. In the picture, there are 6 edges, so 12 is the total degree).

Maximum Walk in a Graph

For a graph of n vertices, there is no longest walk assuming that there is at least one edge in the graph.

Symmetric

If (x,y) is an edge, then (y,x) is also an edge.

Composition of Functions (Example)

If f: R+ → R+, f(x) = x³ and g: R+ → R+, g(x) = x + 2. Then, (f ο g)(x) = f(g(x)) = (x + 2)³ and (g ο f)(x) = g(f(x)) = x³ + 2.

Conditional Statements

If p then q. p → q. Only not true when p is true and q is false. It breaks the contract.

Proof by Cases

In proof by case, we are looking at all the possible cases that might arise in a theorem.

Direct Proof

In the direct proof, we assume the hypothesis p is true and we try to prove that q is true. Thus making p→q true.

Cycle

A circuit of length at least 1 in which no vertex occurs more than once, except the first and last vertices which are the same. For example, ⟨2, 3, 2⟩

Piece Wise Formula

(Solution in Picture) { 2ₙ³ if n is odd. aₙ ={ {5ₙ/2 if n is even.

Logical Symbol and Translations

-∀x∀y M(x,y) - ∃x∃y M(x,y) - ∃x∀y M(x,y) - ∀x∃y M(x,y)

Order of Operations

1. ¬ (not) 2. ∧ (and) 3. ∨ (or) 4. → 5. ↔

Universal Set

A = B if and only if A ⊆ B and B ⊆ A

Sets of Sets

A = { { 1, 2 }, ∅, { 1, 2, 3 }, { 1 } } - { 1, 2 } ∈ A. - 1 is not an element of A, so 1 ∉ A, although { 1 } ∈ A. - Furthermore, { 1 } ⊈ A since 1 ∉ A. - The empty set ∅ is not the same as { ∅ }.

Conjunctive Normal Form (CNF)

A Boolean expression that is a product of sums of literals, d₁ • d₂ • .... • dm where each dⱼ for j ∈ {1, ..., m} is a sum of literals.

Disjunctive Normal Form (DNF)

A Boolean expression that is a sum of products of literals. c₁ + c₂ + .... + cm where each cⱼ for j ∈ {1, ..., m} is a product of literals.

Cartesian Product

A and B, denoted A x B, is the set of all ordered pairs in which the first entry is in A and the second entry is in B. A x B = { (a, b) : a ∈ A and b ∈ B }.

Common Ratio

A geometric sequence is a sequence that increases or decreases by a constant factor. Such as moving up x6, or moving down /2.

Adjacency Matrix

A matrix that records the number of direct links between vertices.

Set Partitions

A non-empty set A is a collection of non-empty subsets of A such that each element of A is in exactly one of the subsets. Requirements: - For all i, Aᵢ ⊆ A. - For all i, Aᵢ ≠ ∅ - A₁, A₂, ...,Aₙ are pairwise disjoint (their intersections are empty). - A = A₁ ∪ A₂ ∪ ... ∪ Aₙ

Prime Number

A number that can only be multiplied by 1 and itself. CAN BE EVEN and ODD. Cannot be negative. Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, etc.

Minterm

A product of literals u₁u₂...uᵢ such that each uⱼ is either vⱼ or vⱼ.

Proof by Contrapositive

A proof that makes use of the fact that p→q is equivalent to its contrapositive ¬q→¬p. So we assume that ¬q is true and try to prove that ¬p is true.

Leaf

A vertex which has no children. (Ex: The leaves are a, f, c, k, i, and j.)

Open walk

A walk in which the first and last vertices are not the same. So, {4, 5, 6, 1, 2}

Closed walk

A walk in which the first and last vertices are the same. So, {4, 5, 6, 1, 4}

Composite Number

A whole number that can be made by multiplying other whole numbers. CAN BE ODD and EVEN. Can be negative. Example: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81,82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99.

Reflexive

All self-loops are present.

Proper Subset

An element of B that is not an element of A (i.e., A ≠ B). A ⊂ B.

Proof by Contradiction

An indirect proof where if the theorem being proven has the form p → q, then the beginning assumption is p ∧ ¬q which is logically equivalent to ¬(p → q).

Trail

An open walk in which no edge occurs more than once.

f: R → R, where f(x) = -x + 3. What is f-1? (Inverse)

The result of solving the equation y = -x + 3 for x is x = -y + 3. Therefore, f-1(y) = -y + 3 which is the same as f-1(x) = -x + 3.

Parents

Can include the root as well. Any branch that has a child.

Isomorphic Graph

Check by seeing if they have: -Same number of vertices -Same number of edges -Same number of vertice degrees.

Post-Order

The root is visited after the children.

Preorder

The root is visited before the children.

Logical Operator

Conjunction — p ∧ q. True only if p and q are true. Disjunction — p ∨ q. True when either of p or q, or both is true. Exclusive or — p ⊕ q. True if either one of p or q is true but not both. Negation — ¬p. Negates value. If True then not True, if False then not False.

Contrapositive

Contrapositive of p → q is ¬q → ¬p.

Converse

Converse of p → q is q → p.

Z

The set of all integers. For example, ..., -2, -1, 0, 1, 2, ...

Common Difference

Each term increases or decreases by the same constant value. Such as moving up +4 or moving down -10.

Trasitive

Edges (e,a) and (a,b) imply the presence of edge (e,b).

Inverse

Inverse of p → q is ¬p → ¬q.

Circuit

Is a closed walk in which no edge occurs more than once. ⟨ 1, 2, 3, 2, 1 ⟩

Path

Is a trail in which no vertex occurs more than once.

Predicate

Logical statement whose truth value is a function of one or more variables.

∴

Means "therefore".

Hasse Diagram

Only used for partial orders. Generally works bottom (lowest number) to top. It will always be in the same direction due to the anti-symmetry property.

N

The set of natural numbers: All integers greater than or equal to 0. For example, 0, 1, 2, ...

Proof by Exhaustion

Prove the statement by checking each element individually. Unlike proof by cases, exhaustive proof looks at every case in a way that is not general.

Power Sets

Set A, denoted P(A), is the set of all subsets of A. if A = { 1, 2, 3 }, then: -- P(A) = { ∅, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }

Depth-First

Starting at a root vertex, scan each neighbor and its neighbors and their neighbors. Once a neighbor has no more neighbors, backtrack to the previous neighbor and exhaust its neighbors.

Ancestor

The ancestors of vertex g are h, d, and b. Every vertex along the path from v to the root (except for the vertex v itself) is an ancestor of vertex v.

Set Cardinality

The cardinality of a finite set A, denoted by |A|, is the number of elements in A. If A = { 2, 4, 6, 10 }, then |A| = 4.

Descendant

The descendants of vertex h are c, g, and k.

Partially Ordered Set

The domain is {3, 5, 6, 7, 10, 14, 20, 30, 60}. x ⪯ y or "x is at most y" if x evenly divides y.

Sibling

They have the same parent. (Ex: Vertices h, i, and j are siblings because they have the same parent, which is vertex d.)

Inverse Finding

To inverse, the equation must be bijective. So, f(x) = y if and only if f-1(y) = x.

Proof by Counterexample

Used to disprove a universal statement. For example, to disprove the statement, "All prime numbers are odd" find one example where this statement is false—the number 2—which is both even and prime.

Bound Variable

Variable is bound to a quantifier. In (∀x P(x)) ∧ Q(x), P(x) is the bound variable.

Free Variable

Variable is free to take on any value in the domain. In (∀x P(x)) ∧ Q(x), Q(x) is the free variable.

Child

Vertices c and g are the children of vertex h.

Breadth-First

Visits vertices in the graph according to their proximity to the start vertex. Starting at a root vertex scan each neighbor before scanning their neighbors.

Rooted Tree

When there is a root and items are set in a particular order.

Free Tree

When there is no particular organization of the vertices and edges.

Maximum Cycle in a Graph

With a graph of n vertices, the length of a cycle can be no longer than n. Example of the picture, since it has 6 vertices, the graph can have no longer than 6.

Maximum Path in a Graph

With a graph of n vertices, the maximum length of a path would be n-1. For example, in the picture of nine vertices, the length of the path is 8: ⟨C, I, F, D, A, E, B, H, G⟩

f(x) = 3x + 1 and g(x) = x². Then, What is (f ο g)(2)? (Compositing)

Work inside out. (f o g)(x) = f(g(x)). Apply g(2) into x² to get 4. Then plugin 4 into 3x + 1 for f(4) which is 13. The answer is 13.

Composition of Functions

f and g are two functions, where f: X → Y and g: Y → Z. The composition of g with f, denoted g ο f, is the function (g ο f): X → Z, such that for all x ∈ X, (g ο f)(x) = g(f(x)).

Sequence

of nth term is 2ⁿ. Solving a sequence depends on the formula. So tₙ = 5ₙ - 4 if n = 1 would be t₁ = 5(1) - 4 = 1.

Biconditional Statement

p if and only if q. p ↔ q. True when p and q have the same truth value (including False ↔ False) and is false when p and q have different truth values.

Logical Equivalence

p is logically equivalent to q. p ≡ q. Logically equivalent if they have the same truth value regardless of the truth values of their individual propositions. So p ≡ ¬¬p or ¬p ≡ p → ¬q.

DeMorgan's Laws for Quantifiers

¬∀ x F(x) as "Not every bird can fly." Which is logically equivalent (≡) to ∃ x ¬ F(x) as "There exists a bird that cannot fly." ¬∃ x Q(x) as "It is not true that there is a child in the class who is absent today." Which is logically equivalent (≡) to ∀ x ¬ Q(x) as "Every child in the class is not absent today."