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}