skip to primary navigationskip to contentDiscrete Mathematics
Proofs, Numbers, and Sets
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Notes
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Lectures
Michaelmas term
Lecture 1:
introduction; proof; implication; modus ponens;
Lecture 2:
modus ponens; bi-implication; divisibility; congruence; universal
quantification; equality; conjunction.
Lecture 3:
existential quantification; unique existence; disjunction.
Lecture 4:
disjunction; Fermat's Little Theorem; reciprocal in modular
arithmetic; negation; proof by contradiction.
Lecture 5:
proof by contrapositive; proof by contradiction; natural numbers;
monoids; commutativity; semirings; cancellation; inverses; integers;
rationals; rings; fields.
Lecture 6:
division theorem; division algorithm; modular arithmetic; integer
linear combinations.
Lecture 7:
integer linear combinations; sets; membership; comprehension; set
equality; sets of common divisors; gcd; Euclid's Algorithm.
Lecture 8:
properties of gcds; Euclid's Theorem; extended Euclid's Algorithm;
Diffie-Hellman cryptographic method: shared secret key, key exchange.
Lecture 9:
mathematical induction; Fundamental Theorem of Arithmetic; Euclid's
infinitude of primes.
Lecture 10:
sets; extensionality; subsets and supersets; separation principle;
Russell's paradox; empty set; cardinality; powerset axiom;
cardinality of powersets.
Lecture 11:
powerset Boolean algebra; pairing axiom; ordered pairs; products.
Lecture 12:
big unions; big intersections; union axiom; disjoint unions.
Lent term
Lecture 13:
relations; internal diagrams; relational extensionality; relational
composition; matrices.
Lecture 14:
relations and matrices; directed graphs; adjacency matrix;
preorders; reflexive-transitive closure.
Lecture 15:
partial functions; functions; inductive definitions.
Lecture 16:
retractions and sections; bijections; equal cardinality; equivalence relations;
set partitions.
Lecture 17:
calculus of bijections; characteristic functions; finite cardinality;
infinity axiom.
Lecture 18:
surjections and injections; axiom of choice; enumerability;
uncountable and unbounded cardinality.
Lecture 19:
direct and inverse images; replacement axiom; set-indexed
constructions; foundation axiom.
Formal Languages and Automata
Exercises
Additional notes
