CMSC-203 Discrete Math: Vocabulary (spring 2001)
Each student is responsible for thoroughly learning
all of the vocabulary items presented in
the lectures and required readings. I hope
that this partial list will help you in this process,
Chapter 5: Sets
- equality
- set, class, element, subset
- membership, containment
- null set (empty set), power set, universal set
- cardinality
- ordinal number, cardinal number, transfinite number
- number, numeral
- ordered pair, ordered tuple
- finite, infinite
- unit, zero
- natural number, integer, rational number, real number, complex number
- irrational number, transcendental number, algebraic number
- sign, magnitude, absolute value
- proper subset, nonempty (nontrivial) subset
- union, intersection, set difference, set complement, Cartesian product
- symmetric difference
- and (conjunction), or (disjunction), not
- Kleene star
- alphabet, language, string, string length, empty string
- lexicographic ordering
- proof, formal proof, informal proof
- proposition, theorem, lemma, corollary, statement
- hypothesis, conjecture, claim, axiom, postulate, supposition, assumption
- direct proof, indirect proof (contradiction), counterexample
- aleph null
- Venn diagram, set notation
- John Venn, Georg Cantor, Giuseppe Peano, Bertrand Russel, David Hilbert,
Kurt Goedel, Alan Turing
- Zermelo Frankel axioms, Peano postulates
- axiom of extension, classification axiom
- De Morgan's Law for sets
- Cartesian coordinates, polar coordinates
- prefix, infix, postfix
- active voice, passive voice, subjunctive voice
- commutative, associative, distributive
- identity, complement
- absorption, idempotent, nilpotent
- prove, show, argue, explain, demonstrate, verify
- because, hence, since, therefore, consequently, thus, it follow that
- although, though, even though, but, yet, albeit, despite
- moreover, furthermore, also, in addition
- Boolean algebra, group, ring, field, semigroup
- disjoint , mutually disjoint (pairwise disjoint)
- partition, covering
- Russel's paradox, halting problem, Loeb 's paradox, Berry's paradox
Chapter 1: Unquantified Logic
- logic, model, truth value
- statement, proposition, expression
- not (negation), or (disjunction), and (conjunction), nand, exclusive-or (xor)
- logical equivalence, contradiction
- informal proof, formal proof, sound/unsound proof, valid/invalid argument, falacy
- true, false, undefined, bottom, logical value
- truth table
- implication, conditional, biconditional
- DeMorgan's Law (of logic)
- falsehood, tautology
- logical laws (see Theorem 1.1.1, p. 14): commutative, associative, distributive, identity, negation, double negation, idempotent, De Morgan, universal bound, absorption, negations of tautology and contradiction
- inverse, converse, contrapositive
- necessary, sufficient
- premise, hypothesis, conclusion
- rules of inference, modus pones, modus tollens
- disjunctive addition, conjunctive simplification, disjunctive syllogism, hypothetical syllogism, case analysis, rule of contradiction
- induction, deduction, abduction
- circuits, series circuit, parallel circuit, gates, sequential/combinatorial circuit, state
- Boolean expression
- recognizer, predicate
- Peirce arrow
- number systems, radix, unary, binary, ternary, decimal, octal, hexadecimal, radix point, two's complement, one's complement
- carry, sum, half-adder, parallel adder
- digital, analog, discrete, continuous
- Aristotle, Immanuel Kant, Augustus De Morgan, Claude Shannon, John Tukey, George Boole
Chapter 2: Quantified Logic
- first-order predicate logic, predicate calculus, propositional calculus
- propositional functions, open sentences, predicate, domain, target
- free variable
- truth set
- universal/existential quantifiers, implicit quantification
- universal/existential statements, universal/existential conditionals
- example, counterexample
- exhaustion
- formal/informal language
- trivial, proper/improper, vacuous
- limit, limit of sequence
- inverse, converse, contrapositive, negation
- Prolog, Maple, Matlab, Mathematica, Macsyma, Latex, Word97, postscript, Ghostview, dvi, ASCII
- universal instatiation
- macro expansion
- universal modus pones, universal modus tollens
- axiom, postulate
- axiom of choice
- converse error, inverse error
- Pythagorean Theorem
- Charles S. Peirce, G. W. Leibniz, Lewis Carroll, Colmerauer and Roussel
Chapter 3: Proofs
- informal/formal proof
- direct proof, indirect proof (contradiction), counterexample, contraposition
- case analysis, applying definitions
- counting argument, diagonalization, zero-knowledge proof, statistical proof
- integers, natural numbers, rational/irrational numbers
- even, odd
- zero, unit, prime, composite
- divides, divisor, proper divisor, non-trivial divisor
- Fundamental Theorem of Arithmetic (unique factorization)
- constructive/nonconstructive proof
- circular logic
- define, type, quantify
- Fermat's Last Theorem
- lemma, corollary, theorem, proposition
- number theory, modular arithmetic
- modulus, remainder
- gcd (greatest common divisor), lcm (least common multiple)
- modular equivalence, mod, floor, ceiling
- sieve of Eratosthenes, factoring, primality testing, trial division
- division theorem (quotient-remainder theorem)
- parity
- triangle inequality
- algorithm, data type, data structure, data abstraction
- Euclid's algorithm (division algorithm)
- Pierre de Fermat, Euclid, Lady Lovelace, Charles Babbage, al-Khowarizmi
Chapter 4: Induction
- Principle of Mathematical Induction (PMI), strong and weak forms
- constructive induction
- inductive set, basis case, inductive step, inductive hypothesis
- implication vs. logical implication
- Well Ordering Principle (WOP)
- sequence, infinite sequence, series, term, index
- special sequences: alternating, constant, arithmetic, geometric, harmonic
- Taylor series, including e, sine, cosine
- summation, product, factorial, logarithm, upper and lower limits
- floor, ceiling
- linearity of summation, change of variable
- closed-form, expanded form, open form, simplification
- existence and uniqueness
- division, prime, unit, zero, composite, greatest common divisor (gcd)
- divison theorem (quotient-remainder theorem), Euclid's Algorithm
- programming language semantics: operational model, denotational semantics, pre- and post-conditions
- loop invariant, guard
- directed graph
- alternating quantifiers, epsilon/delta proof
- recursive definition, recurrence relation
- G. H. Hardy, Joseph L. Lagrange, Anthony Ralston, Bertrand Russell, Edsger Dijkstra
Chapter 7: Functions
- function, relation, (finite, simple) graph
- source, target (co-domain), domain, image (range), rule of assignment
- inverse, inverse image, composition, binary function
- arrow diagram, directed graph, undirected graph, vertices, edges, graph of function
- total function, partial function
- computable, uncomputable
- identity function, constant function, characteristic function, Boolean function, hash function
- injection (one-to-one), surjection (onto), bijection
- endomorphism, permutation, homomorphism, isomorphism, automorphism
- exponential function, logarithm function, change of base rule for logs, natural log, polynomial
- Hamming distance
- projection, restriction
- sequential circuit, combinational circuit
- finite state automaton, state transition function
- language. language accepted by a machine, string, empty string
- mod n operator, equivalence mod n
- additive group modulo n, multiplicative group modulo n
- Euler phi function (totient)
- pigeonhole principle, proof by diagonalization
- finite, infinite, countable, uncountable, uncountably infinite
- cryptographic function, encryption, authentication, RSA, public-key cryptography
- function computed by a program
- lifting an operation, overloading an operation
- Vito Volterra, Peter Dirichlet, Richard Hamming, David Hilbert, Galileo Galilei
Chapter 8: Recursion
- recurrence relation, difference equation, recursive program, recursive process
- initial conditions, order, homogeneous, inhomogeneous, linear
- Fibonacci numbers, Tower of Hanoi, golden ratio
- Stirling numbers (of second kind), Catalan numbers
- guess and check (by induction), iteration, characteristic equations, generating functions
- exact vs. inexact solutions, finite vs. asymptotic analysis
- calculation, simplification, numeric and symbolic computation, approximation
- canonical forms
- Maple, Mathematica, Macsyma, Matlab
- sequences: constant, arithmetic, geometric, alternating, telescoping, harmonic, power
- characteristic equation, difference operator, shift operator
- quadratic equation, fundamental theorem of algebra
- polar coordinates for expressing complex numbers
- even function, odd function, sine, cosine, tangent
- summation, product, summations of common sequences (see above)
- McCarthy function, Ackermann function
- Edouard Lucas, Fibonacci brothers, Douglas Hofstadter, John McCarty, Wilhelm Ackermann
Chapter 10: Relations
- relation, binary relation, k-ary relation, inverse relation
- congruence modulo n
- equivalence relation, relexive, symmetric, transitive
- transitive closure
- total, partial
- antisymmetric
- irreflexive, asymmetric, intransitive
- partition, equivalency class, representative
- group, group of symmetries, dihedral group, reflection, rotation
- finite state automaton
- order relation, strict vs. non-strict order relation, total vs. partial order
- partially ordered sets (posets), chain
- lexicographic order
- Hasse diagram
- minimum, minimal (local minimum)), maximum, maximal (local maximum)
- topological sort, PERT (program evaluation and review technique), CPM (critical path method)
- Ernst Mach, Carl Frederich Gauss,
Chapter 6: Counting (see also 7.4, 7.6, 8.1)
- counting, characteristic function, cardinality
- discrete vs. continuous probability, measure, probability measure, counting measure
- sample space (event space), event, random variable, probability mass function, probability density function
- distribution, cummulative distribution
- mean, mode, median, range, variance, standard deviation, moments
- random, pseudorandom
- countable, uncountable (uncountably infinite), enumeration
- aleph null, aleph one, continum hypothesis
- Baye's Theorem, conditional probability
- fundamental principles of counting: addition and product rules
- Pigeonhole principle, generalized pigeonhole principle, Ramsey theory, infinite pigeonhole principle
- permutation, k-permutation, combination, "n falling k"
- D'Alembert's counting method
- Binomial theorem
- principle of inclusion/exclusion, the "ghastly" formula
- Urn model, ordering vs. no ordering, replacement vs. no replacement
- Pascal's Triangle, Pascal's Formula
- Catalan numbers
- binomial distribution, multinomial distribtution, Bernouli distribution
- Polya's theory of enumeration, Burnside's Lemma
- counting with summations and recurrences
- Cantor's proof, proof by diagonalization, proof by counting argument, probabilistic existence proof
- backgammon, die and dice, pips, cards, poker hands
- Gaussian distribution, central limit theorem
- complexity theory, complexity class
- determinism, nondeterminism, P vs. NP
- reversible computation, Landaurer's principle, Fredkin gate
- quantum computation, Qbit, quantum uncertainty
- Andrei Kolmogorov, Blaise Pascal, Charles Bennett