Discrete Structures Final

Disproving Conjectures Approaches (2 options)

(1) Prove that a conjectures negation is true. (2) Find a counter-example.

Prove S and T are equal (2 methods)

(1) prove both S ⊆ T and T ⊆ S (2) Express w/ set builder, prove, convert back to set notation (difficult)

Fallacies

* Affirming the Conclusion = q -> p, therefore p * Denying the Hypothesis = q -> p, !q, therefore !p * Begging the Question = ..., q, ..., therefore q

Binomial Theorem

**** that

Function

1. Relation from X to Y 2. f(x) is defined \forall x \in X 3. (x,y) \in f, then y is the only value returned by f(x)

Rules of inference

1. `Addition` $p / \therefore p \lor q$ 2. `Simplification` $p \land q / \therefore p$ 3. `Conjunction` $p, q / \therefore p \land q$ 4. `Modus Ponens` $p, p \to q / \therefore q$ 5. `Modus Tollens` $\overline{q}, p \to q / \therefore \overline{p}$ 6. `Hypothetical Syllogism` (aka `Transitivity of Implication`) $p \to q, q \to r / \therefore p \to r$ 7. `Disjunctive Syllogism` (aka `One or the Other`) $p \lor q, \overline{p} / \therefore q$ 8. `Resolution` $p \lor q, \overline{p} \lor r / \therefore q \lor r$

Matrix

A matrix is an n-dimensional collection of values. (rows x columns)

Reflexive (aka Weak) Partial Order

A relation R on set A if it is reflexive, antisymmetric, and transitive.

Irreflexivity (of Relations)

A relation R on set A is irreflexive if \forall a ∈ A, (a,a) ∉ R.

Equivalence Relation

A relation on set A if it is reflexive, symmetric, and transitive.

Partition

A separation of all set members into disjoint subsets.

Total Order

A weak-partially-ordered relation R on set A is a total order if each pair of elements a,b ∈ A is comparable. (Or: antisymmetric, transitive, and comparable)

Proper Subset (Quantification)

A ⊂ B = \forall z (z \in A -> z \in B) \land \exists w (w \notin A \land w \in B)

Subset (Quantification)

A ⊆ B = \forall z (z \in A -> z \in B)

String*

A* set of strings that can be formed using elements of alphabets in a, including lambda A = {a,b} A* = {lambda,a,b,ab,ba,aa,bb,...}

Inverse Relation

All of the ordered pairs reversed in a set; r={(b,a)|(a,b) ∈ R}

Ordered Pair

An ordered pair is a group of two items (a,b) such that (a,b) != (b,a) unless a = b.

Hypothesis/conclusion terms

Antecedent, consequent Hypothesis, conclusion Sufficient, necessary

Combinatorial Proof

Argument based on the principles of counting e.g. Proof (direct, combinatorial): **** if i know

Recursive Definitions

Basis clause - how trivial cases are handled Inductive clause - how complex problem instances in terms of simpler instances External clause - provides bounds on the definition Example: 13,10,7,4,1 Basis: S1=13 Inductive: Sn=Sn-1 - 3 External: Defined for \forall 1 <= n <= 5

Bijective Function (One-to-one Correspondance)

Both inejctive and surjective

Types of Proofs

Direct (only one we need to study this time): Conjecture of the form p -> q Assume p is true, show q is true

Lambda String

Empty string

Surjective Function (Onto)

Every value of Y is used

Injective Function (One-to-one)

Every value of x has a unique value of y

Countably Infinite

Exists a biject mapping between the set and either Z+ or Z*

String

Finite sequence where elements are drawn from a set called the alphabet

Addition Principle

For ordered steps where counts can be moved at one location, use n_1 + n_2 + ... + n_s eg how many ways can 3 characters be printed

Multiplication Principle

For ordered steps where counts can't be moved, use n_1 * n_2 * ... * n_s eg 3 choices on Thur, 4 choices in Fri, and 6 choices on Sat; 3 * 4 * 6 = 72

LHRRWCC

Has the form R(n)=c1*R(n-1) + c2*R(n-2) + ... + ck * R(n-k) Degree is how many previous items it depends on Fibonacci sequence depends on two previous items

Related

If (x,y) ∈ R, x is related to y (xRy)

Increasing vs non-decreasing vs strictly increasing

Increasing = non-incresaing = i_n <= i_(n+1) Strictly increasing = i_n < i_(n+1)

Irreflexive (aka Strict) Partial Order

Irreflexive, antisymmetric, transitive

Join vs Meet vs Boolean Product Operations

Join: A \lor B Meet: A \land B Boolean Product: product using \land

Comparable

Let R be a weak partial order on set A. aka a,b ∈ A and (a,b) ∈ R or (b,a) ∈ R.

Matrix Symmetry

Matrix A is symmetric if A = A^T

r-Permutation

Ordering of an r-element subset of n distinct elements is expressed as P(n,r) n is number items, r is number selected P(n,r)=n!/(n-r)! **order matters**

Composite

Plug in y values of set A into the x values of set B and take the x of A and y of B Think of f∘g = f(g(x))

Proof structure

Proof (<type>): ... Therefore, <restate goal>.

Reflexivity

Relation R on set A is reflexive when (a,a) ∈ R, \forall a ∈ Needs self loop on a graph Needs to have rows along matrix diagonal

Symmetry

Relation R on set A is symmetric if (a,b) ∈ R whenever (b,a) ∈ R, for a,b, ∈ A Cannot have single-direction arrows in graph; all relations should be loops (or self loops) Needs to have lower left triangle mirror the upper right triangle in matrix

Transivity

Relation R on set A is transitive whenever (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R, where a,b,c ∈ R Multiply matrices and check if any 0's were lost

Antisymmetry

Relation r on set A is antisymmetric if (x,y) ∈ R and x != y, then (y,x) ∉ R, \forall x,y ∈ A May have one direction relations and self loops, no full loops

Recurring Relation

Sequence where elements depend on one or more preceding elements f_n=f_(n-1)+f_(n-2)

Proper Subset (General definition) (⊂)

Set A is a proper subset of set B if A ⊆ B and A != B

Subset (General definition) (⊆)

Set A is a subset of set B if every member of A can be found in B

Set Equality

Sets A and B are equal iff A ⊆ B and B ⊆ A.

Cartesian Product

The cartesian product of sets A and B (A x B) is the set of all ordered pairs (a,b), a ∈ A, b ∈ B. Or X x Y = {(x,y) | x ∈ X \land y ∈ Y} |A x B| = |A| * |B|

nth Matrix Power

The nth power of an m x m A(A^n) is the result of n - 1 successive matrix products of A.

Power Set (P(A))

The power set of set A is the set of all of A's subsets, including the empty set. |P(x)| = 2^|x| e.g. A = {a,b,c} P(A) = {Ø, {a}, {b}, {c}, {a,b}, {a,b}, {b,c}, {a,b,c}}

Pascals Triangle

Triangle with ones on the edges where each element is the sum of the two elements above it

Disjoint

Two sets are disjoint if their intersection is the empty set. e.g. C = {a,e,i,o,u} and D={g,j,p,q,y} C ∩ D = Ø, thus C and D are disjoint

Free (a.k.a. Unbound) variable

Unquantified variables

Matrix Product

Use the rows of the position in matrix A and the columns of the position in matrix B. Multiply the values and add them all together.

Weak vs Strong Induction

Weak assumes truth of a single preceding case Strong assumes truth of all preceding cases

Repetition and Combination of Combinations

When repetition is allowed, the number of r-combinations in a set is (n+r-1 r)=(n+r-1 n-1) When repetition is allowed and one element of each group is included, then (r-1 r-n)=(r-1 n-1)

Express X - Y in Logic

X - Y = {z | z ∈ X \land Z \∉ Y}

f: X -> Y f(n) = p [ (n,p) \in f ] Domain, codomain, maps, image, pre-image, range

X is the domain of f Y is the codomain of f f maps X to Y p is the image of n n is the pre-image of p f's rage is the set of all images of X's elements

(Binary) Relation

X to set Y is a subset of the Cartesian product of X (the domain) and Y (the codomain).

Express X ∩ Y in Logic

X ∩ Y = {z | z ∈ X \land z ∈ Y}

Express X ∪ Y in Logic

X ∪ Y = {z | z ∈ X \lor z \in Y}

Basic sets

Z = all integers Z+, N+ = positive integers Z*, N0 = non-negative integers Z- = negative integers Zeven = even integers Zodd = odd integers Q = rational numbers R = real numbers

Proof by Contradiction

\lnot (p -> q) = p \land \lnot q

Generalized De Morgan's Laws

\overline{\forall x P(x)} \equiv \exists x \overline{P(x)}

Well-Formed Formula (wff)

a correctly structured expression of a language

Fallacy

argument constructed with an improper inference

Inductive Argument

argument that moves from specific observations to a general conclusion

Deductive Argument

argument that uses accepted general principles to explain a specific situation

(Logically) Equivalent (p ≡ q)

both evaluate to the same result when presented with the same input

compound proposition

claim that is a logical combination of multiple simple propositions

proposition (a.k.a. statement)

claim that is either true or false with respect to an associated context

Domain of Discourse (a.k.a. Universe of Discourse)

collection of values from which a variable's value is drawn

Theorem

conjecture that has been shown to be true

Argument

connected series of statements to establish a definite proposition

Arithmetic Sequence

d = a_(n+1) - a_n

Valid argument

deductive argument in the form of (p1 and p2 and p3 and ...) -> q

Prolog: defining relations

e.g. ancestor(X,Y) :- parent(X,Y) ancestor(X,Y) :- parent(Z,Y), ancestor(X,Z)

Recursive Algorithm

express the solution of a task in terms of a simpler case of the same problem

Affine Transformations

http://u.arizona.edu/~mccann/classes/245/slides/20180302-_files/20180302_007.png

Pidgeonhole Principle

n items in k boxes means each box contains at least ceil(n/k) items aka

rth Boolean Power

n x n 0-1 matrix A(A^[r]) is the n x n matrix resulting from r - 1 successive boolean products. A^[0] = I_n

Proof by Contraposition

p -> q = \lnot q -> \lnot p

Inverse

p \to q is \lnot p \lnot q

Contrapositive

p \to q is \lnot q \to \lnot p

Converse

p \to q is q \to p

simple proposition

proposition containing no logical operators

Contradiction

proposition that always evaluates to false

Tautology

proposition that always evaluates to true

Contingency

proposition that is neither a tautology nor a contradiction

Bound variable

quantified variable in a predicate

Geometric Sequence

r = g_(n+1) / g_n

r-Combinations

r-Combination of an n-element set X is an r-element subset of X is expressed as C(n,r) n is number items, r is number selected C(n,r)=(n r)=n!/(r! * (n-r)!) **order doesn't matter**

Lemma

simple theorem whose truth is used to construct more complex theorems (opposed to corollary)

Proof

sound argument that establishes the truth of a theorem

Predicate (a.k.a. Propositional Function)

statement that includes at least one variable and will evaluate to either true or false when the variable(s) are assigned value(s)

Conjecture

statement with an unknown truth value

Philosophical Logic

the classical notion of 'logic': The study of thought and reasoning, including arguments and proof techniques

Discrete Mathematics

the study of collections of distinct objects

Corollary

theorem whose truth follows directly from another theorem (opposed to lemma)

Mathematical Logic

use of formal languages and grammars to represent the syntax and semantics of computation

Sound argument

valid argument that also has true premises

Principle of Inclusion-Exclusion

x of group 1 and y of group 2, but z overlap There are x + y - z choices

Express \lnot X in Logic

{z | z ∉ X}