Discrete Structures
섹션 개요
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Diskretne strukture VSŠ 63705
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Predavanja: Matematična indukcija. Izjave, izjavni vezniki.
Lectures: Mathematical induction (pages 5-7); Introduction into propositional logic: atoms, logical connectives, logical formulas, truth tables, equivalence of formulas (pages 95-99).
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Predavanja: Izjavni izrazi, resničnostna tabela, enakovredni izjavni izrazi. Zakoni izjavnega računa. DNO in KNO. Polni nabori izjavnih veznikov.
Lectures: Laws of propositional logic (pages 11-15); Disjunctive and conjuctive normal form (pages 4-6); Functionally complete sets of connectives
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Predavanja: Sklepanje v izjavnem računu. Formalizacija sklepov, definicija pravilnega sklepa, pravila sklepanja, dokaz pravilnosti sklepa. Pomožni sklepi - pogojni sklep, sklep s protislovjem, analiza primerov.
Lectures: Logical inference - formalization, rules of inferece, derivation of rules (pages 100-104). Proof by contradiction (section 8 - on this link you can freely download the book), proof by conditional derivation (section 6.2), proof by cases (explained at the bottom of section 3.2).
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Predikatni račun - jezik predikatnega računa, izjavne formule. Doseg kvantifikatorjev, proste in vezane spremenljivke. Interpretacija izjavne formule, enakovrednost izjavnih formul, preimenovanje spremenljivk, zakoni predikatnega računa,
Predicate logic: predicates; universe of discourse, universal and existential quantifier. (link) Scope of quantifiers. Free and bound variables. Well-formed formulas. Interpretation of formulas, equivalence of formulas. Laws of predicate calculus.
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Zakoni predikatnega računa z omejitvami. Preneksna normalna oblika.
Množice, osnovne operacije z množicami ($\cap, \cup, +, \setminus, ^c$), lastnosti komplementiranja, osnovne enakosti množic.
Laws of predicate calculus wrt. naming of variables. Prenex normal form.
Sets, basic operations with sets ($\cap, \cup, +, \setminus, ^c$), laws for operations with sets. (notes, on this link there are some proofs of laws of set theory).
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Definicija relacije, grafična predstavitev relacij, lastnosti relacij, operacije z relacijami, potence relacij, tranzitivna in tranzitivno-refleksivna ovojnica relacije.
Lectures: Relations, definition and examples, graphical representation (link to last two topics), properties of relations (link), operations with relations (link), transitive closure and reflexive-transitive closure (pages 28-30)
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Predavanja: Preslikave, njihove lastnosti, kompozitum preslikav, inverzna preslikava. Lastnosti kompozituma preslikav. Ekvivalenčna relacija, ekvivalenčni razredi, faktorska množica. Nekaj malega o neskončnih množicah.
Lectures: Mappings (link), properties (link), inverse mapping (link), composition of functions (link). Properties of a composition of functions (link). Equivalence relation (link), equivalence classes, factor set. (link) A taste of infinite sets.
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Predavanja: Teorija grafov - lema o rokovanju, grafična zaporedja, družine grafov (polni in prazni grafi, polni dvodelni grafi, cikli, poti), podgrafi, izomorfizem grafov.
Lectures: Graph theory - definition of a graph (link), degree of a vertex, handshaking lemma (Theorem 5.1.1), graphical sequences (link), isomorphism of graphs (link), families of graphs (complete and empty graphs, complete bipartite graphs, cycles, paths) (link) , subgraphs. (link)
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Predavanja: Sprehod, pot, obhod, cikel. Povezanost in komponente za povezanost. Razdalja v grafu. Karakterizacija dvodelnih grafov. Eulerjev problem. Drevesa in gozdovi.
Lectures: Walk, path , closed walk, cycle. (link) Connected graphs and components. Distance in graphs. (link) Characterization of bipartite graphs. (link) Euler's problem (link). Trees, forests (link). -
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Predavanja: ta teden ni pouka.
Lectures: no lectures/exercises this week.
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Predavanja: Permutacije. Zapis s tabelicami in zapis z disjunktnimi cikli.
Lectures: Permutations (link). Tables and disjoint cycles representations.
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Predavanja: Parnost permutacij, igra 15. Potenčne enačbe.
Lectures: Parity of permutaions, 15 puzzle. Equations with permutation powers.
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Ta teden letos - zaradi uskladitve števila petkovih terminov - predavanja pri DSvsp odpadejo.
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