Discrete Math from Rosen
Unordered pair
A set with two members
Conjunction (rule of inference)
((p) ∧ (q)) → (p ∧ q)
Hypothetical syllogism
((p → q) ∧ (q → r)) → (p → r)
Resolution (rule of inference)
((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r)
Disjunctive syllogism
((p ∨ q)∧¬p) → q
Exclusive or
(aka Exclusive Disjunction) True when at MOST one of the two propositions is true.
Inclusive or
(aka Inclusive Disjunction) True when at LEAST one of the two propositions is true.
Hypothesis
(aka antecedent or premise )The first portion of an implication
Conclusion
(aka consequence) The second portion of an implication.
Basis (matrices)
A basis for a subspace H of Rn is a linearly independent set in H that spans H.
Equivalence relation
A binary relation that is reflexive, symmetric, and transitive
Directed graph
A collection of points and arrows connecting some pairs of points
Graph
A collection of points and lines connecting some pairs of points
Contradiction
A compound proposition that is always false
Solution
An assignment of truth values that makes a compound proposition true
Element
An object in a set
String
A finite list of symbols from an alphabet
Alphabet
A finite, nonempty set of objects called symbols
Indirect proof
Proofs of theorems that are not direct proofs; i.e. they do not start with the premises and end with the conclusion
argument
a sequence of statements
arithmetic progression
a sequence of the form a, a + d, a + 2d, . . . , where a and d are real numbers
geometric progression
a sequence of the form a, ar, ar2, . . . , where a and r are real numbers
countable set
a set that either is finite or can be placed in one-to-one correspondence with the set of positive integers
uncountable set
a set that is not countable
infinite set
a set that is not finite
finite set
a set with n elements, where n is a nonnegative integer
lemma
a theorem used to prove other theorems
rule of inference
a valid argument form that can be used in the demonstration that arguments are valid
a ≡ b (mod m) (a is congruent to b modulo m)
a − b is divisible by m
greedy algorithm
an algorithm that makes the best choice at each step according to some specified condition
linear combination of a and b with integer coefficients
an expression of the form sa + tb, where s and t are integers
inverse of a modulo m
an integer a such that aa ≡ 1 (mod m)
composite
an integer greater than 1 that is not prime
prime
an integer greater than 1 with exactly two positive integer divisors
primitive root of a prime p
an integer r in Zp such that every integer not divisible by p is congruent modulo p to a power of r
fallacy
an invalid argument form often used incorrectly as a rule of inference (or sometimes, more generally, an incorrect argument)
element, member of a set
an object in a set
modular arithmetic
arithmetic done modulo an integer m ≥ 2
zero-one matrix
matrix with each entry equal to either 0 or 1
fundamental theorem of arithmetic
Every positive integer can be written uniquely as the product of primes, where the prime factors are written in order of increasing size.
RSA cryptosystem
the cryptosystem where P and C are both Z26, K is the set of pairs k = (n, e) where n = pq where p and q are large primes and e is a positive integer, Ek(p) = pe mod n, and Dk(c) = cd mod n where d is the inverse of e modulo (p − 1)(q − 1)
f (x) is \theta{(g(x))}
the fact that f (x) is bothO(g(x)) and(g(x))
f(x) is O(g(x))
the fact that |f (x)| ≤ C|g(x)| for all x > k for some constants C and k
conclusion
the final statement in an argument or argument form
f ◦ g (composition of f and g)
the function that assigns f (g(x)) to x
inverse of f
the function that reverses the correspondence given by f (when f is a bijection)
worst-case time complexity
the greatest amount of time required for an algorithm to solve a problem of a given size
discrete logarithm of a to the base r modulo p
the integer e with 0 ≤ e ≤ p − 1 such that re ≡ a (mod p)
x (floor function)
the largest integer not exceeding x
gcd(a, b) (greatest common divisor of a and b)
the largest integer that divides both a and b
|S| (the cardinality of S)
the number of elements in S
searching algorithm
the problem of locating an element in a list
encryption
the process of making a message secret
Universal instantiation
the rule of inference used to conclude that P(c) is true, where c is a particular member of the domain, given the premise ∀xP(x). Universal instantiation is used when we conclude from the statement "All women are wise" that "Lisa is wise," where Lisa is a member of the domain of all women.
domain of f
the set A, where f is a function from A to B
codomain of f
the set B, where f is a function from A to B
universal set
the set containing all objects under consideration
A ∪ B (the union of A and B)
the set containing those elements that are in at least one of A and B
A ∩ B (the intersection of A and B)
the set containing those elements that are in both A and B
A' (the complement of A)
the set of elements in the universal set that are not in A
continuum hypothesis
the statement there no set A exists such that ℵ0 < |A| < c
Mathematical Induction
the statement ∀n P(n) is true if P(1) is true and ∀k[P(k) → P(k + 1)] is true. Cannot be used to find new theorems, or explain why a particular theorem is true.
\sum_{i=1}^{n} a_{i}
the sum a_1 + a_2 + ... a_n
domain of discourse
the values a variable in a propositional function may take
a | b (a divides b)
there is an integer c such that b = ac
De Morgan's Laws
¬(p ∧ q) ≡ ¬p ∨¬q, and ¬(p ∨ q) ≡ ¬p ∧¬q
Double negation law
¬(¬p) ≡ p
Lemma
A less important theorem that is helpful in the proof of other results. Example: the pumping lemma
Edge
A line in a graph
k-tuple
A list of k objects
Sequence
A list of objects
Ordered pair
A list of two elements
Zero-one matrix
A matrix all of whose entries are either 0 or 1.
Vector (matrices)
A matrix with only one column is called a column vector, or simply a vector
Square Matrix
A matrix with the same number of rows and columns
Symbol
A member of an alphabet
Cycle
A path that starts and ends in the same node
Simple path
A path without repetition
Node
A point in a graph
Vertex
A point in a graph
Property
A predicate
Relation
A predicate, most typically when the domain is a set of k-tuples
sieve of Eratosthenes
A procedure for finding all primes not exceeding a specified number n, described in Section 4.3
Trivial proof
A proof which uses the fact that the conclusion is true
Vacuous proof
A proof which uses the fact that the hypothesis is false
Irrational number
A real number that is not rational.
Rational number
A real number where there exist integers p and q with q != 0 such that r=p/q.
Matrix
A rectangular array of numbers. Plural is matrices. By convention, represented as A, B, C, etc.. with 1..j..m columns and 1..i..n rows. A particular cell in a matrix is denoted by a_{ij}, b_{kn}, c_{lo}, etc
Binary Relation
A relation whose domain is a set of pairs
Path
A sequence of nodes in a graph connected by edges
Argument
A sequence of statements that end with a conclusion
Argument
A sequence of statements that end with a conclusion.
Language
A set of strings
Singleton set
A set with one member
Symmetric matrix
A square matrix where A = A^t. Thus A = [aij ] is symmetric if aij = aji for all i and j with 1 ≤ i ≤ n and 1 ≤ j ≤ n. In other words, transposing the matrix doesn't change the matrix.
Linear independence
An indexed set of vectors fv1; : : : ; vpg in Rn is said to be linearly independent if the vector equation x1v1 C x2v2 C C xpvp D 0 has only the trivial solution.
Argument (of a function)
An input to a function
Member
An object in a set
Boolean operation
An operation on Boolean values
Bit operations
An operation on a bit or bits. Correspond to logical connectives, i.e. OR=∨, AND=∧, and XOR=⊕
Complement
An operation on a set, forming the set of all elements not present
Union
An operation on sets combining all elements into a single set
Cartesian product
An operation on sets forming a set of all tuples of elements from respective sets
Intersection
An operation on sets forming the set of common elements
Concatenation
An operation that joins strings together
Function
An operation that translates inputs into outputs
Tautology
A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it
Satisfiable
A compound proposition where there's at least one assignment of truth values to its variables that makes it true
Direct proof
A conditional statement p->q constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. Lead from the premises to the conclusion.
Tree
A connected graph without simple cycles
Proposition
A declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. e.g. Washington, DC is the capital of the United States of America; or 1+1=2; but NOT x + 1 = 2.
Set
A group of objects
Transposition (matrices)
Denoted by A^t, is the n × m matrix obtained by interchanging the rows and columns of A. In other words, if A^t = [b_ij], then b_ij = a_ji for i = 1, 2, . . . , n and j = 1, 2, . . . , m. Basically, every row i becomes a column j, and every column j becomes a row i.
Bézout's theorem
If a and b are positive integers, then gcd(a, b) is a linear combination of a and b.
Truth value
If it is a true proposition, denoted by T and the truth value of a proposition is false, denoted by F, if it is a false proposition
Fermat's little theorem
If p is prime and p | a, then ap−1 ≡ 1 (mod p).
Premises (of an argument)
In the context of an argument, the statements leading to a conclusion.
Predicate
Refers to a property that the subject of the statement can have. Also a function whose range is {TRUE, FALSE}.
S = T (set equality)
S and T have the same elements
S ⊂ T (S is a proper subset of T)
S is a subset of T and S = T
axiom
a statement that is assumed to be true and that can be used as a basis for proving theorems
Conjecture
a statement that is being proposed to be a true statement, usually on the basis of some partial evidence, a heuristic argument, or the intuition of an expert
premise
a statement, in an argument, or argument form, other than the final one
empty string
a string of length zero
Identity laws
p ∧ T ≡ p; p ∨ F ≡ p
Absorption laws
p ∨ (p ∧ q) ≡ p; p ∧ (p ∨ q) ≡ p
Distributive laws
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r); p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Domination laws
p ∨ T ≡ T; p ∧ F ≡ F
Idempotent laws
p ∨ p ≡ p; p ∧ p ≡ p
Commutative laws
p ∨ q ≡ q ∨ p; p ∧ q ≡ q ∧ p
Negation laws
p ∨¬p ≡ T; p ∧¬p ≡ F
predicate
part of a sentence that attributes a property to the subject
scope of a quantifier
portion of a statement where the quantifier binds its variable
brute force
the algorithmic paradigm based on constructing algorithms for solving problems in a naive manner from the statement of the problem and definitions
space complexity
the amount of space in computer memory required for an algorithm to solve a problem
time complexity
the amount of time required for an algorithm to solve a problem
Inductive Hypothesis
the assumption that P(k) is true
average-case time complexity
the average amount of time required for an algorithm to solve a problem of a given size
hexadecimal representation
the base 16 representation of an integer
binary representation
the base 2 representation of an integer
octal representation
the base 8 representation of an integer
n = (akak−1 . . . a1a0)b
the base b representation of n
ℵ0 (aleph null)
the cardinality of a countable set
c
the cardinality of the set of real numbers
Converse
the conditional statement q → p. Does NOT have the same truth value as p->q for all possible truth values of p and q. e.g. Converse of "If it is raining, then the home team wins" is "If the home team wins, then it is raining"
Inverse
the conditional statement ¬p →¬q. Does NOT have the same truth value as p->q for all possible truth values of p and q. e.g. Inverse of "If it is raining, then the home team wins" is "If it isn't raining, then the home team doesn't win"
Contrapositive
the conditional statement ¬q →¬p. Always has the same truth value as p -> q. e.g. Contrapositive of "If it is raining, then the home team wins" is "If the home team doesn't win, then it isn't raining"
f (x) is \omega{(g(x))}
the fact that |f (x)| ≥ C|g(x)| for all x > k for some positive constants C and k
At (transpose of A)
the matrix obtained fromAby interchanging the rows and columns
In (identity matrix of order n)
the n × n matrix that has entries equal to 1 on its diagonal and 0s elsewhere
set builder sotation
the notation that describes a set by stating a property an element must have to be a member
cryptanalysis
the process of recovering the plaintext from ciphertext without knowledge of the encryption method, or with knowledge of the encryption method, but not the key
decryption
the process of returning a secret message to its original form
Basis Step
the proof of P(1) in a proof by mathematical induction of ∀nP (n)
Inductive Step
the proof of P(k) → P(k + 1) for all positive integers k in a proof by mathematical induction of ∀nP (n)
Exclusive Or
the proposition "p XOR q," which is true when exactly one of p and q is true, but not both. E.g. this entree comes with a choice of soup or salad. Denoted by p ⊕ q
Conjunction
the proposition "p and q," which is true if and only if both p and q are true, and is false otherwise.
Disjunction
the proposition "p or q," which is false when both p and q are false and is true otherwise
universal quantification
the proposition ∀xP(x) that is true if and only if P(x) is true for every x in the domain
existential quantification
the proposition ∃x P(x) that is true if and only if there exists an x in the domain such that P(x) is true
a mod b
the remainder when the integer a is divided by the positive integer b
sorting
the reordering of the elements of a list into prescribed order
Existential instantiation
the rule of inference that allows us to conclude that there is an element c in the domain for which P(c) is true if we know that ∃xP(x) is true. We cannot select an arbitrary value of c here, but rather it must be a c for which P(c) is true. Usually we have no knowledge of what c is, only that it exists. Because it exists, we may give it a name (c) and continue our argument. "Something loves to wag its tail. c loves to wag his tail"
Universal generalization
the rule of inference that states that ∀xP(x) is true, given the premise that P(c) is true for all elements c in the domain. Universal generalization is used when we show that ∀xP(x) is true by taking an arbitrary element c from the domain and showing that P(c) is true. "Any particular dog loves to wag its tail. Therefore, all dogs love to wag their tails."
A ⊕ B (the symmetric difference of A and B)
the set containing those elements in exactly one of A and B
A − B (the difference of A and B)
the set containing those elements that are in A but not in B
P(S) (the power set of S)
the set of all subsets of S
range of f
the set of images of f
∅ (empty set, null set)
the set with no members
lcm(a,b)
the smallest positive integer that is divisible by both a and b
x (ceiling function)
the smallest integer greater than or equal to x
M recognizes A
aka M accepts A
circular reasoning
(aka begging the question) reasoning where one or more steps are based on the truth of the statement being proved
Theorem
(aka facts or results) A statement that can be shown to be true.
Implication
(aka material implication) A conditional statement, consisting of a hypothesis , followed by a conclusion. Often expressed as p->q. Statement only false if the hypothesis is true and the conclusion is false. If the conclusion is true, the truth of the hypothesis doesn't matter. Helpful to think of as a contractual obligation that you expect to be fulfilled. Example: If I am elected, then I will lower taxes.
Axiom
(aka postulates) Statements we assume to be true, the premises, if any, of the theorem, and previously proven theorems.
Propositional calculus
(aka propositional logic) The area of logic that deals with propositions
Propositional Variables
(aka statement variables) variables that represent propositions, just as letters are used to denote numerical variables. By convention p,q,r,s... are used.
linear and binary search algorithms
(given in Section 3.1). Linear = O(n) worst case, Binary = O(n log n) worst case
Truth Value
(of a proposition) This value is true, denoted by T, if it is a true proposition, or value is false, denoted by F, if it is a false proposition.
Modus ponens
(p ∧ (p → q)) → q
Simplification (rule of inference)
(p ∧ q) → p
Associative laws
(p ∨ q) ∨ r ≡ p ∨ (q ∨ r); (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
Modus tollens
(¬q ∧ (p → q))→¬p
Subspace (matrices)
A subspace of Rn is any set H in Rn that has three properties: a. The zero vector is in H. b. For each u and v in H, the sum u C v is in H. c. For each u in H and each scalar c, the vector cu is in H. g
Bit
A symbol with two possible values, namely 0 and 1. Can be used to represent a truth value, customarily 1= T and 0 = F.
Chinese remainder theorem
A system of linear congruences modulo pairwise relatively prime integers has a unique solution modulo the product of these moduli.
Truth table
A table with a row for each of the possible truth values of a proposition p. Each row shows the truth value of ¬p corresponding to the truth value of p for this row.
Corollary
A theorem that can be established directly from a theorem that has been proved
Proof
A valid argument that establishes the truth of a mathematical statement. Used to demonstrate that theorems are true.
Boolean variable
A variable that is either true or false. Can be represented using a bit.
Validity (of an argument)
The conclusion must follow from the truth of the premises. An argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false.
Biconditional
The proposition "p if and only if q". Denoted p <-> q. Can also be expressed "p is necessary and sufficient for q" or "p iff q". Example: If and only if I am elected, I will lower taxes.
Negation
The proposition with truth value opposite to the truth value of p. Read "not p". In English, "It is not the case that p"
onto function, surjection
a function from A to B such that every element of B is the image of some element in A
one-to-one function, injection
a function such that the images of elements in its domain are distinct
one-to-one correspondence, bijection
a function that is both one-to-one and onto
Connected graph
Boolean AND operation
Parity
The even-ness or odd-ness of a number; All even numbers share the same parity, as do all odd numbers
Conclusion (of an argument)
The final statement of an argument
sequence
a function with domain that is a subset of the set of integers
algorithmic paradigm
a general approach for constructing algorithms based on a particular concept
Venn diagram
a graphical representation of a set or sets
Bit string
a list of bits
Logic gate
a logic element that performs a logical operation on one or more bits to produce an output bit
paradox
a logical inconsistency
valid argument form
a sequence of compound propositions involving propositional variables where the truth of all the premises implies the truth of the conclusion
argument form
a sequence of compound propositions involving propositional variables
Identity Matrix
The identity matrix of order n is the n × n matrix In = [δ_ij ], where δ_ij = 1 if i = j and δ_ij = 0 if i != j . Multiplying a matrix by an appropriately sized identity matrix does not change this matrix.
Well-ordering property
Appendix 1
pairwise relatively prime integers
a set of integers with the property that every pair of these integers is relatively prime
Universal modus ponens
Combines universal instantiation with modus ponens ∀x(P(x) → Q(x)) P(a), where a is a particular element in the domain ∴ Q(a) Example: Every truck that is a firetruck is red. I see a firetruck. Therefore, it must be red.
Universal modus tollens
Combines universal instantiation with modus tollens ∀x(P(x) → Q(x)) ¬Q(a), where a is a particular element in the domain ∴ ¬P(a) Example: Every truck that is a firetruck is red. I see a blue truck. Therefore, it cannot be a firetruck.
Fallacies
Common forms of incorrect reasoning, which lead to invalid arguments
Dimension (matrices)
Given a nonzero subspace H, denoted by dimH, is the number of vectors in any basis for H. The dimension of the zero subspace f0g is defined to be zero.2
Scalar (matrices)
Given a vector u and a real number c, the scalar multiple of u by c is the vector cu obtained by multiplying each entry in u by c. The number c in cu is called a scalar; it is written in lightface type to distinguish it from the boldface vector u.
Meet (matrix)
Given zero-one matrices A and B, the zero-one matrix with (i, j )th entry aij ∧ bij . Denoted by A ∧ B.
Join (matrix)
Given zero-one matrices A and B, the zero-one matrix with (i, j )th entry aij ∨ bij. Denoted by A ∨ B. Basically you OR every bit.
Summation (matrices)
Let A = [a_ij] and B = [b_ij] be m × n matrices. Given the matrices being summed have equal size (rows and columns), the sum of A and B, denoted B, is the m x n matrix that has a_{ij} + b_{ij} as its (i, j)th element. Alternatively, A+B = [a_{ij} + b_{ij}]
Boolean product (matrix)
Let A = [aij ] be an m × k zero-one matrix and B = [bij ] be a k × n zero-one matrix. Then A and B, denoted by AB, is the m × n matrix with (i, j)th entry c_ij where: c_ij = (a_i1 ∧ b_1j ) ∨ (a_i2 ∧ b_2j ) ∨ · · · ∨ (a_ik ∧ b_kj).
Product (matrices)
Let A be an m x k matrix and B be a k x n matrix. The product of A and B, denoted by AB, is the m x n matrix with its (i, j)th entry equal to the sum of the products of the corresponding elements from the ith row of A and jth column of B. In other words, if AB = [c_ij], then: c_ij = a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{ik}b_{kj}. Important note: The product of two matrices is not defined when the number of columns in the first matrix and the number of rows in the second matrix are not the same. Another important note: matrix multiplication is not commutative (AB != BA)
division algorithm
Let a and d be integers with d positive. Then there are unique integers q and r with 0 ≤ r < d such that a = dq + r.
Compound Propositions
New propositions that are formed from existing propositions using logical operators
Compound propositions
New propositions that are formed from existing propositions using logical operators.
bitwise XOR
The bitwise XOR of two strings is the string obtains by taking the XOR of the corresponding bits from both strings
Logical Order of Operations
Parens, Neg, Conj, Disj, Cond, Bicond
Proof by contraposition
Take !q as a premise, and using axioms, definitions, and previously proven theorems, together with rules of inference, we show that !p must follow.
Rank (matrices)
The A, denoted by rank A, is the dimension of the column space of A.
Valid
The conclusion must follow from the truth of the preceding statements
Propositional logic
The area of logic that deals with propositions
bitwise AND
The bitwise AND of two strings is the string obtains by taking the AND of the corresponding bits from both strings
bitwise OR
The bitwise OR of two strings is the string obtains by taking the OR of the corresponding bits from both strings
Existential generalization
The rule of inference that is used to conclude that ∃xP(x) is true when a particular element c with P(c) true is known. That is, if we know one element c in the domain for which P(c) is true, then we know that ∃xP(x) is true. "Rover loves to wag his tail. Therefore, something loves to wag its tail."
Range
The set from which outputs of a function are drawn
Domain
The set of possible inputs to a function
Empty set
The set with no members
Propositional function
The statement P(X)
Empty string
The string of length zero
Element (matrices)
The value in the a_{ij} cell
Boolean value
The values TRUE or FALSE, often represented by 1 or 0
Zero vector
The vector whose entries are all zero is called the zero vector and is denoted by 0. (The number of entries in 0 will be clear from the context.)
Uniqueness quantifier
There exists exactly one unique X such that P(x) is true
insertion sort
a sorting that at the j th step inserts the j th element into the correct position in in the list, when the first j − 1 elements of the list are already sorted
conjecture
a mathematical assertion proposed to be true, but that has not been proved
theorem
a mathematical assertion that can be shown to be true
Unsatisfiable
When a compound proposition is false for ALL possible assignments of truth values to its variables
Equivalent propositions
When two compound propositions always have the same truth value, e.g. a conditional statement and its contrapositive, or it's converse and inverse.
Equality (matrices)
When two matrices have the same number of rows and columns and the corresponding entries in every position are equal.
bubble sort
a sorting that uses passes where successive items are interchanged if they in the wrong order
propositional function
a statement containing one or more variables that becomes a proposition when each of its variables is assigned a value or is bound by a quantifier
block cipher
a cipher that encrypts blocks of characters of a fixed size
character cipher
a cipher that encrypts characters one by one
affine cipher
a cipher that encrypts the plaintext letter p as (ap + b) mod m for integers a and b with gcd(a, 26) = 1
shift cipher
a cipher that encrypts the plaintext letter p as (p + k) mod m for an integer k
set
a collection of distinct objects
pseudoprime to the base b
a composite integer n such that bn−1 ≡ 1 (mod n)
satisfiable compound proposition
a compound proposition for which there is an assignment of truth values to its variables that makes it true
Contingency
a compound proposition that is sometimes true and sometimes false
linear congruence
a congruence of the form ax ≡ b (mod m), where x is an integer variable
proof
a demonstration that a theorem is true
recurrence relation
a equation that expresses the nth term an of a sequence in terms of one or more of the previous terms of the sequence for all integers n greater than a particular integer
string
a finite sequence
algorithm
a finite sequence of precise instructions for performing a computation or solving a problem
cryptosystem
a five-tuple (P, C,K, E,D) where P is the set of plaintext messages, C is the set of ciphertext messages, K is the set of keys, E is the set of encryption functions, and D is the set of decryption functions
Affirming the consequent
a formal fallacy of inferring the converse from the original statement; In other words, treating an argument with premises p → q and q and conclusion p as a valid argument form, which it is not. "If it is snowing, then it is below 32F" "It is below 32F", therefore "it is snowing"
Denying the antecedent
a formal fallacy of inferring the inverse from the original statement; In other words, treating an argument with the premises p → q and ¬p and conclusion ¬q as a valid argument form, which it is not. "If it is snowing, then is it below 32F" "It is not snowing", therefore "it is not below 32F"
uncomputable function
a function for which no computer program in a programming language exists that finds its values
computable function
a function for which there is a computer program in some programming language that finds its values
symmetric matrix
a matrix is symmetric if it equals its transpose
digital signature
a method that a recipient can use to determine that the purported sender of a message actually sent the message
roster method
a method that describes a set by listing its elements
rational number
a number that can be expressed as the ratio of two integers p and q such that q = 0
witness to the relationship f (x) is O(g(x))
a pair C and k such that |f (x)| ≤ C|g(x)| whenever x > k
Mersenne prime
a prime of the form 2p − 1, wherep is prime
intractable problem
a problem for which no worst-case polynomial-time algorithm exists for solving it
tractable problem
a problem for which there is a worst-case polynomial-time algorithm that solves it
solvable problem
a problem that can be solved by an algorithm
unsolvable problem
a problem that cannot be solved by an algorithm
linear search algorithm
a procedure for searching a list element by element
binary search algorithm
a procedure for searching an ordered list by successively splitting the list in half
proof by cases
a proof broken into separate cases, where these cases cover all possibilities
Cantor diagonalization argument
a proof technique used to show that the set of real numbers is uncountable
nonconstructive existence proof
a proof that an element with a specified property exists that does not explicitly find such an element
constructive existence proof
a proof that an element with a specified property exists that explicitly finds such an element
exhaustive proof
a proof that establishes a result by checking a list of all possible cases
proof by contradiction
a proof that p is true based on the truth of the conditional statement ¬p → q, where q is a contradiction
vacuous proof
a proof that p → q is true based on the fact that p is false
trivial proof
a proof that p → q is true based on the fact that q is true
proof by contraposition
a proof that p → q is true that proceeds by showing that p must be false when q is false
direct proof
a proof that p → q is true that proceeds by showing that q must be true when p is true
uniqueness proof
a proof that there is exactly one element satisfying a specified property
corollary
a proposition that can be proved as a consequence of a theorem that has just been proved
key exchange protocol
a protocol used for two parties to generate a shared key
matrix
a rectangular array of numbers
Logic circuit
a switching circuit made up of logic gates that produces one or more output bits
membership table
a table displaying the membership of elements in sets
encryption key
a value that determines which of a family of encryption functions is to be used
free variable
a variable not bound in a propositional function
bound variable
a variable that is quantified
counterexample
an element x such that P(x) is false
nondeterministic finite automaton
aka NFA
logic circuit
aka digital circuit; receives input signals p1, p2, . . . , pn, each a bit [either 0 (off) or 1 (on)], and produces output signals s1, s2, . . . , sn, each a bit
accept state
aka final states
finite state machine
aka finite automaton
Propositional variables
aka statement variables; a variable that represents a proposition
valid argument
an argument with a valid argument form
function from A to B
an assignment of exactly one element of B to each element of A
partial function
an assignment to each element in a subset of the domain a unique element in the codomain
without loss of generality
an assumption in a proof thatmakes it possible to prove a theorem by reducing the number of cases to consider in the proof
relatively prime integers
integers a and b such that gcd(a, b) = 1
private key encryption
encryption where both encryption keys and decryption keys must be kept secret
public key encryption
encryption where encryption keys are public knowledge, but decryption keys are kept secret
Bézout coefficients of a and b
integers s and t such that the Bézout identity holds
b is the image of a under f
b = f(a)
gates
basic logic circuits, AND, OR, and NOT
S ⊆ T (S is a subset of T)
every element of S is also an element of T
logically equivalent expressions
expressions that have the same truth value no matter which propositional functions and domains are used
Carmichael number
composite integer n such that n is a pseudoprime to the base b for all positive integers b with gcd(b, n) = 1
Consistent Compound Propositions
compound propositions for which there is an assignment of truth values to the variables that makes all these propositions true
logically equivalent compound propositions
compound propositions that always have the same truth values
negation operator
constructs a new proposition from a single existing proposition
a is a pre-image of b under f
f(a) = b
Euclidean algorithm
for finding greatest common divisors by successively using the division algorithm (see Algorithm 1 in Section 4.3)
computational model
idealized computer
connectives
logical operators used to form new propositions from two or more existing propositions
a and b are congruent modulo m
m divides a − b
Addition (rule of inference)
p → (p ∨ q)
Bitwise operations
operations on bit strings that operate on each bit in one string and the corresponding bit in the other string
Logical operators
operators used to combine propositions
Bézout identity
sa + tb = gcd(a, b)
matrix addition
see page 178
matrix multiplication
see page 179
cardinality
two sets A and B have the same cardinality if there is a one-to-one correspondence from A to B
