du03.workbook

{ "versionNumber": 2, "cells": [ { "id": 1, "type": "text", "data": "::::header\n:::hgroup\n::p[Logika pre informatikov]{.h2.text-muted}\n# 3. domáca úloha\n:::\n\n:::div{.dl-indented}\n:icon[deadline] Termín odovzdania\n: Výnimočne kvôli veľkonočným sviatkom utorok 7. 4. 2026 o 23:59:59\n\n:icon[bonus] Hodnotenie\n: Čisto informatívne, nezarátava sa do hodnotenia predmetu.\n: Odovzdané úlohy ohodnotíme slovne a percentuálne.\n: Využite získanú spätnú väzbu na prípravu na semestrálne testy.\n\n:icon[diagram-2-fill]{.text-primary} Vetva repozitára\n: du03\n\n:icon[save-fill]{.text-success} Uloženie riešenia do vetvy\n: Stlačte :span[Merge changes]{.border.border-success.text-success.p-1.rounded} na hornej lište.\n: Určite uložte posledné zmeny pred odovzdaním!\n\n:icon[submission] Odovzdanie\n: Stlačte :span[:icon[send-fill]{.text-info} Hand In]{.border.border-info.text-info.rounded.p-1.text-nowrap} na hornej lište.\n: Ak by automatické odovzdanie zlyhalo, vytvorte pull request (PR)\n - z vetvy du03 v repozitári ‹váš_github_login›/lpi26-‹váš_uniba_login›\n - do vetvy du03 v repozitári FMFI-UK-1-AIN-412/lpi26-‹váš_uniba_login›.\n: Všetky ďalšie zmeny hárku uložené pomocou :span[Merge changes]{.text-success.text-nowrap} sa pridajú do PR.\n: PR vytvorte iba raz (či už ručne alebo pomocou :span[Hand in]{.text-info.text-nowrap}).\n\n:icon[send-check-fill]{.text-info} Kontrola odovzdania\n: Správne vytvorený PR nájdete v zozname PR pre du03.\n\n:icon[help] Návody\n: Vytvorenie PR: náš (trocha zastaralý) návod, dokumentácia GitHubu.\n: Použitie aplikácie Logic Workbook: na konci hárka pre tc01.\n\n:icon[envelope-at-fill]{.text-primary} Kontakt na vyučujúcich\n: lpi-team :icon[at] lists.dai.fmph.uniba.sk\n:::\n::::\n::::details\n::summary[:icon[help] Tabuľka symbolov používaných nižšie a ich $\TeX$ového zápisu]\n:::div{.table.small.table-sm.overflow-auto}\n| Symbol | TeX | Význam | Symbol | TeX | Význam | Symbol | TeX | Význam |\n|:------:|:----|:-------|:------:|:----|:-------|:------:|:----|:-------|\n| ∧ | \\land | konjunkcia | $\Lang$ | \\Lang | jazyk | $\models$ | \\models | pravdivosť v štruktúre |\n| ∨ | \\lor | disjunkcia | $\Atoms$ | \\Atoms | množina atómov jazyka $\Lang$ | $\nmodels$ | \\nmodels | nepravdivosť v štruktúre |\n| → | \\limpl | implikácia | $\Consts$ | \\Consts | množina konštánt jazyka $\Lang$ | $\pmodels$ | \\pmodels | pravdivosť v ohodnotení (propozičná) |\n| ↔ | \\lequiv | ekvivalencia | $\Preds$ | \\Preds | množina predikátov jazyka $\Lang$ | $\npmodels$ | \\npmodels | nepravdivosť v ohodnotení |\n| ≐ | \\doteq | rovnosť (ako logický) symbol | $\Forms$ | \\Forms | množina všetkých formúl jazyka $\Lang$ | $\pentails$ | \\pentails | výrokovologické (propozičné) vyplývanie |\n| ¬ | \\lnot | negácia | $\Struct$ | \\Struct | štruktúra | $\npentails$ | \\npentails | výrokovologické nevyplývanie |\n| ∀ | \\forall | všeobecný kvantifikátor | $\sym{Ann}$, $\asym{2}{likes}$ | \\sym{Ann}, \\asym{2}{likes} | konkrétny mimolog. symbol | ↦ | \\mapsto | zobrazenie (prvku na prvok) |\n| ∃ | \\exists | existenčný kvantifikátor | $\vk$, …, $\vz$ | \\vk, …, \\vz | konkrétna indivíduová premenná | $\Tabl$ | \\Tabl | metapremenná ozn. tablo |\n| $\sign T$, $\sign F$ | \\sign{T}, \\sign{F} | znamienko označenej formuly | $\bigl($, $\bigr)$| \\bigl(, \\bigr) | väčšie zátvorky | $\pi$ | \\pi | metapremenná ozn. vetvu tabla |\n:::\n::::", "comments": [] }, { "id": 3, "type": "text", "data": "## Úloha 3.:cnt[a]\n\n:small[:icon[star]{.text-primary} Hodnota: 70 % · :icon[source] Zbierka: :xref[3.4.5]{.exbook label="príklad"}.]\n\nUvažujme jazyk $\Lang$ výrokovologickej časti logiky prvého rádu bez rovnosti.\nNech $X$ a $Y$ sú ľubovoľné výrokovologické formuly jazyka $\Lang$, nech $T$ je ľubovoľná\nvýrokovologická teória v $\Lang$.\n\nDokážte alebo vyvráťte na základe definícií vyplývania a pravdivosti vo výrokovologickom ohodnotení:", "comments": [] }, { "id": 5, "type": "text", "data": "::h3[a)]{.h6.mt-0.me-2.mb-0.float-start.lh-base}\n\nAk $T \npentails (X \land Y)$,\ntak $T \pentails \neg X$ alebo $T \pentails \neg Y$.\n\n::h4[:icon[write] Riešenie]{.h6}\n:::aside{.note.bg-transparent.p-0}\n\* Nehodiace sa zmažte.\n:::\n\nTvrdenie :mark[je/nie je]\* pravdivé.\n\n_:mark[Dôkaz/Kontrapríklad]\._\n…\n", "comments": [] }, { "id": 4, "type": "text", "data": "::h3[b)]{.h6.mt-0.me-2.mb-0.float-start.lh-base}\n\nAk $T \pentails X$ alebo $T \pentails Y$,\ntak $T \pentails (X \lor Y)$.\n\n::h4[:icon[write] Riešenie]{.h6}\n:::aside{.note.bg-transparent.p-0}\n\ Nehodiace sa zmažte.\n:::\n\nTvrdenie :mark[je/nie je]\* pravdivé.\n\n_:mark[Dôkaz/Kontrapríklad]\*.\n…\n", "comments": [] } ], "settings": { "katexMacros": "\newcommand{\DeclareMathOperator}[2]{\newcommand{#1}{\mathop{\mathrm{#2}}}}\n\n%% General math\n\n% Domain and range\n\DeclareMathOperator{\dom}{dom}\n\DeclareMathOperator{\rng}{rng}\n\n% Powerset\n\newcommand\powerset[1]{\mathcal{P}(#1)}\n\n% Cardinality\n\newcommand{\card}[1]{\lvert#1\rvert}\n\n% Numbers\n\newcommand{\Nat}{\mathbb{N}}\n\n% Operations\n\newcommand{\TIMES}{\cdot}\n\n% Sets\n\newcommand{\sete}[1]{\{#1\}} % enumerated\n\newcommand{\setc}[1]{\{\,#1\,\}} % comprehended\n\n\n% Meta syntax %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\DeclareMathOperator{\arity}{ar}\n\newcommand{\Lang}{\mathcal{L}}\n\newcommand{\Vars}{\mathcal{V}{\Lang}}\n\newcommand{\Consts}{\mathcal{C}{\Lang}}\n\newcommand{\Preds}{\mathcal{P}{\Lang}}\n\newcommand{\Funcs}{\mathcal{F}{\Lang}}\n\newcommand{\Terms}{\mathcal{T}{\Lang}}\n\newcommand{\Atoms}{\mathcal{A}{\Lang}}\n\newcommand{\Forms}{\mathcal{E}{\Lang}}\n\newcommand{\PAtoms}{\mathcal{PA}{\Lang}}\n\newcommand{\PForms}{\mathcal{PE}{\Lang}}\n\n% Equality axioms\n\newcommand{\Eq}{\mathrm{Eq}}\n\n% Syntactic transformation\n\newcommand{\transform}{\rightsquigarrow}\n\n% Various syntactic functions\n\DeclareMathOperator{\vars}{vars}\n\DeclareMathOperator{\atoms}{atoms}\n\DeclareMathOperator{\acnt}{acnt}\n\DeclareMathOperator{\termVars}{termVars}\n\DeclareMathOperator{\free}{free}\n\DeclareMathOperator{\ground}{ground}\n\DeclareMathOperator{\mgu}{mgu}\n\DeclareMathOperator{\vcount}{vcount}\n\DeclareMathOperator{\acount}{acount}\n\DeclareMathOperator{\ccount}{ccount}\n\DeclareMathOperator{\pcount}{pcount}\n\DeclareMathOperator{\ncount}{ncount}\n\DeclareMathOperator{\cjcount}{cjcount}\n\DeclareMathOperator{\bccount}{bccount}\n\DeclareMathOperator{\lpcount}{lpcount}\n\DeclareMathOperator{\rpcount}{rpcount}\n\DeclareMathOperator{\subfs}{subfs}\n\DeclareMathOperator{\cons}{cons}\n\DeclareMathOperator{\nnf}{nnf}\n\n\n% Concrete syntax %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n% Connectives\n\n\newcommand{\nrarr}{\nrightarrow}\n\newcommand{\limpl}{\rarr}\n\newcommand{\nlimpl}{\nrarr}\n\newcommand{\lequiv}{\lrarr}\n\newcommand{\bigland}{\bigwedge}\n\newcommand{\biglor}{\bigvee}\n\newcommand{\lnand}{\mathbin{\uparrow}}\n\newcommand{\lnor}{\mathbin{\downarrow}}\n\newcommand{\lxor}{\veebar}\n\newcommand{\landnot}{\nrightarrow}\n\newcommand{\emptyclause}{\Box}\n\newcommand{\quant}[2]{\mathop{#1#2}\nolimits}\n\newcommand{\A}{\quant\forall}\n\newcommand{\E}{\quant\exists}\n\n% Symbols\n\newcommand{\sym}[1]{\text{\textsf{#1}}}\n\newcommand{\asym}[2]{\text{\textsf{#2}$^#1$}}\n\newcommand{\var}[1]{\text{\textsf{\textit{#1}}}}\n\newcommand\vk{\var{k}}\n\newcommand\vl{\var{l}}\n\newcommand\vm{\var{m}}\n\newcommand\vn{\var{n}}\n\newcommand\vo{\var{o}}\n\newcommand\vp{\var{p}}\n\newcommand\vq{\var{q}}\n\newcommand\vr{\var{r}}\n\newcommand\vs{\var{s}}\n\newcommand\vt{\var{t}}\n\newcommand\vu{\var{u}}\n\newcommand\vv{\var{v}}\n\newcommand\vw{\var{w}}\n\newcommand\vx{\var{x}}\n\newcommand\vy{\var{y}}\n\newcommand\vz{\var{z}}\n\n\n% Semantics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n% Auxiliary\n\providecommand{\NModels}\n {\mathrel{\mkern1.5mu{\not}\nobreak\mkern-1.5mu}\models}\n\n% General\n\providecommand{\nmodels}{\mathrel{\NModels}}\n\providecommand{\Equiv}{\mathrel{\Leftrightarrow}}\n\renewcommand{\Equiv}{\mathrel{\Leftrightarrow}}\n\newcommand{\entails}{\vDash}\n\newcommand{\nentails}{\nvDash}\n\n% Propositional\n\newcommand{\pmodels}{\models_{\mathrm{p}}}\n\newcommand{\npmodels}{\mathrel{\NModels_{\mathrm{p}}}}\n\newcommand{\pEquiv}{\Equiv_{\mathrm{p}}}\n\newcommand{\pentails}{\entails_{\mathrm{p}}}\n\newcommand{\npentails}{\nentails_{\mathrm{p}}}\n\n% First-order\n\newcommand{\Struct}{\mathcal{M}}\n\newcommand{\InStruct}{^{\mathcal{M}}}\n\n% Tableaux %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\newcommand{\Tabl}{\mathcal{T}}\n\n\newcommand\sign[1]{\mathop{\text{\textsf{\textbf{#1}}}}\nolimits}", "github": { "editBranch": "du03", "handinBranch": "du03" } } }