01 - MTH 231 - Elements of Discrete Mathematics

What is the contrapositive of p -> q ? Of "If it is raining then there are clouds in the sky."?

<image> "If there are no clouds in the sky then it is not raining."

What does "If p then q" imply? Alternate ways of stating this? What is its notation? What is the proper way to state this? What are the terms for p and q in this context? What does the truth table look like? Examples?

In order for the sentence to be true -> if p is true, then q must be true The sentence is false if p is true and q is false. The sentence is true when p is false, regardless of whether q is true (vacuously true, true by default) (Heuristic == p dot contained within q circle) <see image of truth table> q if p q whenever p p only if q p implies q p -> q p => q This is called "the conditional of q by p" or the "implication of q by p". p == hypothesis or antecedent q == conclusion or consequent "It is raining only if there are clouds in the sky." - expresses the same conditional as "If it is raining then there are clouds in the sky." - but not the same as "If there are clouds in the sky then it is raining."

Compound proposition

Propositions that are manipulated or combined using logical operations

Determine whether the function f from {a, b, c, d} to {1, 2, 3, 4, 5} with f(a) = 4, f(b) = 5, f(c) = 1, and f(d) = 3 is one-to-one. f(x) = x2 from the set of integers to the set of integers?

The function f is one-to-one because f takes on different values at the four elements of its domain. The function f(x) = x2 is not one-to-one because, for instance, f(1) = f(−1) = 1, but 1 ≠ −1.

roster method of displaying a set name Z composed of a, b, c, d

Z = {a, b, c, d}

ordered n-tuple

collection that has a₁, a₂, ..., aₙ

What is the contrapositive of: p -> q How is it equivalent?

p -> q ≡ ¬q -> ¬p

What are De Morgan's Laws?

¬(p ∨ q) ≡ p ∧ q ¬(p ∧ q) ≡ p ∨ q

Notation for statement: For every positive real number x there exists a positive real number y such that xy = 1

∀x ∊ (0, inf), ∃y ∊ (0, inf), xy = 1

set

- discrete structure used to group objects together - unordered collection of distinct objects called elements or members of the set i.e. a ∈ A a is an element of set A a ∉ A a is not an element of set A

cardinality of ⎢∅⎢ ?

0 no elements

What is a necessary condition of p? (p -> q) In terms of a hypothesis and its conclusion? In terms of "If x is a non-zero whole number, then x² is a positive whole number."?

A condition that must be satisfied given p is true The conclusion of a true conditional statement is a necessary condition of the hypothesis. "x² is a positive whole number" is a necessary condition of "x is a non-zero whole number". "x is a non-zero whole number" is NOT a necessary condition of "x² is a positive whole number". e.g. x = √2 x² = 2

What is a sufficient condition of q? (p -> q) In terms of a hypothesis and its conclusion? In terms of "If x is a non-zero whole number, then x² is a positive whole number."?

A condition that suffices to guarantee q is true The hypothesis of a true conditional statement is a sufficient condition for the conclusion. "x is a non-zero whole number" is a sufficient condition for "x² is a positive whole number". "x² is a positive whole number" is NOT a sufficient condition for "x is a non-zero whole number". e.g. x² = 2 x = √2

Definition, notation, "if and only if" statement of the proper subset A of B

A is a subset of B but A ≠ B A ⊂ B

Cartesian relation

A subset R of the Cartesian product A × B is called a relation from the set A to the set B. The elements of R are ordered pairs, where the first element belongs to A and the second to B. e.g. R = {(a, 0), (a, 1), (a, 3), (b, 1), (b, 2), (c, 0), (c, 3)} is a relation from the set {a, b, c} to the set {0, 1, 2, 3}, and a relation from the set {a, b, c, d, e} to the set {0, 1, 3, 4).

finite set

Set with n distinct elements where n is a nonnegative integer

surjection

p. 151

power of a set

the set of all subsets of a set (S) e.g. power of set S P(S)

∀x ∈ S(P(x)) meaning? Alternate notation?

universal quantification of P(x) overall elements in the set S ∀x(x ∈ S → P(x))

Notation for: "One of your tools (x) is not in the correct place ( ¬P(x) ) but is in excellent condition ( Q(x) )."

∃x(¬P(x) ∧ Q(x))

Express ∃xP(x) and ∀xP(x) using conjunctions, disjunctions.

∃xP(x) P(1) ∨ P(2) ∨ ... P(xₙ) ∀xP(x) P(1) ∧ P(2) ∧ ... P(xₙ)

What is a biconditional proposition in terms of p, q? Notation?

"p if and only if q" "p iff q" p ⇔ q p ↔ q (p -> q) ∧ (q -> p)

What are some of the core topics covered in this discrete mathematics course? (9)

- logic - set theory - functions - direct and indirect proofs - contradiction and contraposition - induction and recursion - combinatorics - graph theory - spanning trees

What is the inverse of p -> q? Of "If it is raining then there are clouds in the sky."?

<image> "If it is not raining then there are no clouds in the sky."

Increasing, strictly increasing, decreasing, strictly decreasing functions

A function that is either strictly increasing or strictly decreasing must be one-to-one. A function that is increasing, but not strictly increasing, or decreasing, but not strictly decreasing, is not one-to-one. Domain: real numbers x < y and x and y are in the domain of f increasing if f(x) ≤ f(y) strictly increasing if f(x) < f(y) decreasing if f(x) ≥ f(y) strictly decreasing if f(x) > f(y) i.e. increasing if ∀x∀y(x < y → f(x) ≤ f(y)) strictly increasing if ∀x∀y(x < y → f(x) < f(y)) decreasing if ∀x∀y(x < y → f(x) ≥ f(y)) strictly decreasing if ∀x∀y(x < y → f(x) > f(y))

What do subset and superset mean

A is a subset of B if all elements of A are contained within B A ⊆ B i.e. A ⊆ B if and only if ∀x(x ∈ A → x ∈ B) is true. B is a superset of A B ⊇ A * a subset can be a subset of itself e.g. all people in China ⊆ all people in China

Predicate

A property othat a subject of a statement can have e.g. "x is greater than 3" variable == x predicate == "is greater than 3" P(x) i.e. P(x) is the value of the propositional function P at x.

Proposition

A sentence that is true or false proposition: "Oregon is a state in the US." not proposition: - commands, questions "This statement is false."

Bit string

A sequence of zero or more bits of a certain "length" e.g. 010110 bit string of length 6

Universal quantification? Its notation for P(x)? Term for a false element in the statement?

A statement that asserts that a property is true for all values of a variable in a particular domain called the domain of discourse or just the domain ∀xP(x) ∀ - universal quantifier "for all xP(x)" - for every, all of, for each, given any, for arbitrary, for any - best not to use "for any" as it is ambiguous e.g. There is not any reason to avoid studying. There is an x for which P(x) is false. "counterexample" - element for which P(x) is false

Existential quantification? English statement? Notation? It's false means... ?

A statement that is true if and only if at least one value of x is true in the domain Terms: "There exists an element x in the domain such that P(x)." - for some, for at least one, there is ∃xP(x) ∃ == existential quantifier P(x) is false for every x.

function domain codomain image preimage range

Assigns to each element of first set exactly one element of a second set where the two sets are not necessarily distinct A function f from A to B is an assignment of exactly one element of B to each element of A. We write f(a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f : A → B. A is the domain of f and B is the codomain of f. If f(a) = b, we say that b is the image of a and a is a preimage of b. The range, or image, of f is the set of all images of elements of A. Also, if f is a function from A to B, we say that f maps A to B. Note that the codomain of a function from A to B is the set of all possible values of such a function (that is, all elements of B), and the range is the set of all values of f(a) for a ∈ A, and is always a subset of the codomain. That is, the codomain is the set of possible values of the function and the range is the set of all elements of the codomain that are achieved as the value of f for at least one element of the domain. Functions are sometimes also called mappings or transformations.

Set identities table

Basically the same as the logical equivalences table

Provide a useful equivalence to ¬(p -> q) using De Morgan's Laws.

De Morgan's Laws: --------- ¬(p ∨ q) ≡ p ∧ q ¬(p ∧ q) ≡ p ∨ q ---------- (1) Find an equivalence to p -> q that is a conjunction or a disjunction. p -> q is only false when: - p is true and q is false. - or when ¬p is false and q is false So the disjunctive equivalence of the implication is: p -> q ≡ ¬p V q ¬(p -> q) ≡ ¬(¬p V q) ≡ p ∧ ¬q

onto function

Every codomain value is also a range value i.e. Every member of codomain is an image of the function

What does Theorem 1 of set theory state?

Every nonempty set S is guaranteed to have at least two subsets - the empty set and the set S itself ∅ ⊆ S and S ⊆ S

Disjunction of proposition p, q

False when both p, q are false True otherwise (p or q) symbol: p V q

Truth table

Gives the truth of a compound proposition based upon all possible true values of input propositions

What makes a compound proposition a tautology? A contradiction?

If it is true regardless of what the truth values of the propositions it is made up of are (p -> p) V q If it is false regardless of what the truth values of the propositions it is made up of are (p ∧ q) ∧ ¬q

What would make system specifications consistent in terms of their logical expressions?

If the truth table has a row with all "True" in it.

When is a compound proposition satisfiable? Unsatisfiable?

If there is an assignment of truth values to its variables that makes it true (i.e. the assignment is a solution) When it is false for all assignemtns of truth values to its variables or its negation is true for all assignments (.i.e. a tautology)

What does the following statement mean: if x>0 then x := x+1

If x is greater than 0, the x is incremented by 1.

Quantification

Method of creating a proposition from a propositional function Expresses the extent to which a predicate is true over a range of elements terms = all, some, many, none, few predicate calculus - the area of logic that deals with predicates and quantifiers

What are the equivalent statements for: ∃x, (~P(x)) Equivalent notation?

Not all x satisfy P(x) Not each ... Not every ... ~(∀x, P(x))

Equivalent statements for ∀x, (~P(x)) Equivalent notation?

Not any x satisfies P(x) No x... ~(∃x, P(x))

What does it mean for compound propositions P and Q, made up of propositions p₁ ... pₙ, to be logically equivalent? Notation?

P and Q have the same truth values for all possible combinations of truth values for input propositions p₁ ... pₙ . i.e. They have the same truth table given the same arrangement of the truth values for input propositions p₁ ... pₙ . P ≡ Q e.g. ¬(¬p) ≡ p p ∨ q ≡ q ∨ p p ∧ q ≡ q ∧ p

How to obtain the bit string for the union and intersection of two sets? For the union? For the intersection?

Perform bitwise Boolean operations on them union -> 11, 10 == 1 00 == 0 bitwise OR intersection -> 11 == 1, 01, 00 == 0 bitwise AND

Propositional function

Sentence that contains a finite number of variables that becomes a proposition when specific values are substituted for the variables

Difference of A and B sets? (A - B)

Set containing those elements that are in A but not in B A − B = A ∩ B' i.e. the complement of B with respect to A e.g. The difference of {1, 3, 5} and {1, 2, 3} is the set {5}

Union of the sets A and B meaning and notation

Set that contains elements that are either in A or B or both A ∪ B e.g. The union of the sets {1, 3, 5} and {1, 2, 3} is the set {1, 2, 3, 5}; that is, {1, 3, 5}∪{1, 2, 3} = {1, 2, 3, 5}.

Disjoint sets

Sets that do not have any elements in common e.g. Let A = {1, 3, 5, 7, 9} and B = (2, 4, 6, 8, 10}. Because A ∩ B = ∅, A and B are disjoint.

Purpose, components of a venn diagram

Show the relationships between sets - universal set - objects inside the universal set (i.e. sets) e.g. U == all letters in alphabet V == vowels

f(x) = x² from R+ to R+ Increasing, strictly decreasing, ... ?

Strictly increasing Suppose that x and y are positive real numbers with x < y. Multiplying both sides of this inequality by x gives x² < xy. Similarly, multiplying both sides by y gives xy < y². Hence, f(x) = x² < xy < y² = f(y). However, the function f(x) = x² from R to the set of nonnegative real numbers is not strictly increasing because −1 < 0, but f(−1) = (−1)2 = 1 is not less than f(0) = 02 = 0.

What is Discrete Mathematics?

Study of mathematical structures that are typically discrete and granular in nature rather than continuous

Venn diagram of A − B = A ∩ B'

The complement of B is everything except B, including the space around it in U, so that leaves the part of A that doesn't intersect with B.

Intersection of the sets A and B meaning and notation

The set containing those elements in both A and B A ∩ B

Let A = {a, b, c, d, e} and B = {1, 2, 3, 4} with f(a) = 2, f(b) = 1, f(c) = 4, f(d) = 1, and f(e) = 1. What is the image of S ={b, c, d} ?

The set f(S) = {1, 4}

Uniqueness quantification? English statement? Notation?

There exists a unique element x such that P(x) is true - see above "There is exactly one" "There is one and only one" ∃!xP(x) or ∃₁xP(x)

Use De Morgan's Laws for quantifiers to negate the following statements: "Every student in your class has taken a course in calculus." ∀xP(x) "There is a student in your class who has taken a course in calculus." ∃xP(x)

There is a student in your class who has not taken a course in calculus. ∃x¬P(x) or ¬∀xP(x) There is not a student in your class who has taken calculus. ¬∃xP(x) or ∀x¬P(x)

Determine if statement is true and show it: ∃x ∊ R, (x>1 -> ( x²/(x+1) <= 1/2 )

True x = 0 Hypothesis is false, implication is true by default (vacuously true)

Conjunction of proposition p, q

True when both p, q are true False otherwise (p and q) symbol: p ∧ q

Negation of a proposition p

True when p is false and vice versa (not p) symbol: ¬p ~p

predicate logic

Used to express meaning of a wide range of statements in mathematics and computer science in ways that permit us to reason and explore relationships between objects

When are ordered n-tuple collections equal? Aka?

We say that two ordered n-tuples are equal if and only if each corresponding pair of their elements is equal. In other words, (a1, a2, ..., an) = (b1, b2, ..., bn) if and only if aᵢ = bᵢ, for i = 1, 2, ..., n. Ordered pairs

set builder notation of set Z composed of all odd positive itegers less than 10 x

Z = {x ∈ Z⁺ | x is odd and x < 10}

One-to-one function

a function where each element of the range is paired with exactly one element of the domain a.k.a. injective, an injection Function is one-to-one if and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. ∀a∀b(f(a) = f(b) → a = b) or equivalently ∀a∀b(a ≠ b → f(a) ≠ f(b))

Examples of discrete structures built from sets

combinations (unordered collections of objects) relations (sets of ordered pairs representing relationships) graphs (sets of vertices and edges that connect vertices) finite state machines (model computing machines)

Complement of set A

complement of A with respect to the universal set U U - A' or A with line over it e.g. A = {a, e, i, o, u} A' = other letters

Cartesian product of sets A and B

denoted by A × B, is the set of all ordered pairs (a, b), where a ∈ A and b ∈ B

Cartesian notation A²

denotes A × A, the Cartesian product of the set A with itself. Similarly, A3 = A × A × A, A4 = A × A × A × A, and so on e.g. A = {1, 2} A² = {(1, 1), (1, 2), (2, 1), (2, 2)} A³ = {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}.

Domain and codomain of int func (float);

domain -> set of real numbers (float) codomain -> returned integers

Terms, notations for a set with no elements

empty set null set ∅ {}

∃x ∈ S(P(x)) meaning? Alternate notation?

existential quantification of P(x) over all elements in S ∃x(x ∈ S ∧ P(x))

Let R be the relation with ordered pairs (Abdul, 22), (Brenda, 24), (Carla, 21), (Desire, 22), (Eddie, 24), and (Felicia, 22). Here each pair consists of a graduate student and this student's age. Specify a function determined by this relation.

f(Abdul) = 22, f(Brenda) = 24, f(Carla) = 21, f(Desire) = 22, f(Eddie) = 24, and f(Felicia) = 22 domain: {Abdul, Brenda, Carla, Desire, Eddie, Felicia} codomain: {10, ... 100} range: {21, 22, 24}

multiset notation?

multiple member set unordered collection of element that can occur more than once {m1 · a1, m2 · a2, ..., mr · ar} denotes the multiset with element a1 occurring m1 times, ... - mi, i = 1, 2, ..., r, are called the multiplicities

naive vs axiomatic set theory

naive - objects in set not specified, intuitive notion of object, leads to paradoxes axiomatic - incorporating axioms into set theory to avoid paradoxes

Cardinality of a union of two finite sets A and B? Term?

principle of inclusion-exclusion

In the statement P(x₁, x₂, x₃, ... xₙ), what are the terms (3) for P?

propositional function n-place predicate n-ary predicate

What is the converse of p -> q? Of "If it is raining then there are clouds in the sky."?

q -> p "If there are clouds in the sky then it is raining."

real-valued vs integer-valued

real-valued -> codomain is real numbers integer-valued -> codomain is integers * two functions both of the same kind can be added or multiplied

sequence

represent ordered lists of elements

Term for set with one element

singleton set * {∅} is a singleton set, not a null set

cardinality

size of a finite set i.e. n is the cardinality of set S ⎢S⎢ * comes from using the term cardinal number as size of finite set

How do multiplicities work in multisets with unions and intersections? Difference and sum?

unions -> take the largest multiplicity intersection -> take the smallest difference -> multiplicity - multiplicity, if negative it equals 0 sum -> add multiplicities

What is the notation for a proposition that is true iff P(x) is true for every x in domain D? i.e. for every x in D... What is a term for this kind of statement?

∀x ∊ D, P(x) or ∀x, P(x) ∀ = universal quantifier (for all, for every) If P(x) is false for at least one x in D, then statement ∀x, P(x) is false universally quantified statement

Alternate statements using logical symbols and their English equivalents for: ∀x < 0(x² > 0) ∀y ≠ 0(y² > 0) ∃z > 0(z² > 0)

∀x(x < 0 → x² > 0) "The square of a negative real number is positive." ∀y(y ≠ 0 → y² > 0) "The cube of every nonzero real number is nonzero." ∃z(z > 0 ∧ z² > 0) "There is a positive square root of 2."

What is the notation for two sets A, B being equal?

∀x(x ∈ A ↔ x ∈ B) i.e. they are equal if and only if they have the same elements, regardless of the order {1, 3, 5} == {5, 3, 1}

Notation for the statement there exists x in D such that P(x), i.e. is true for at least one x in domain D Term for this kind of statement?

∃x ∊ D, P(x) or ∃x, P(x) ∃ = for some, there exists True if P(x) is true for particular x False if every x for P(X) is false existentially quantified statement