tc02.workbook

{ "versionNumber": 2, "cells": [ { "id": 11, "type": "text", "data": "::::header\n:::hgroup\n::p[Logika pre informatikov]{.h2.text-muted}\n# 2. teoretické cvičenie\n:::\n\n:::div{.dl-indented}\n:icon[diagram-2-fill]{.text-primary} Vetva repozitára\n: tc02\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: Uložte svoje riešenie pred zavretím karty prehliadača alebo odchodom na inú stránku!\n\n:icon[help] Návod\n: Krátky návod na použitie aplikácie Logic Workbook nájdete na konci hárku 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:icon[person-raised-hand]{.text-primary} Konzultačné hodiny\n: Streda o 12:20–13:00 v I-9\n:::\n::::\n::::details\n::summary[:icon[help] :b[Tabuľka symbolov] používaných nižšie a ich $\TeX$ového zápisu]\n\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 $\Lang$ | $\bigl($, $\bigr)$| \\bigl(, \\bigr) | väčšie zátvorky |\n| ¬ | \\lnot | negácia | $\Struct$ | \\Struct | štruktúra | | | |\n| ↦ | \\mapsto | zobrazenie (prvku na prvok) | $\sym{Ann}$, $\asym{2}{likes}$ | \\sym{Ann}, \\asym{2}{likes} | konkrétny mimolog. symbol | | | |\n:::\n::::", "comments": [] }, { "id": 20, "type": "text", "data": "## :icon[homework] Domáca úloha\n\n:icon[bonus] Ďalšie skúsenosti a spätnú väzbu môžete získať vyriešením domácej úlohy:\n\n::::div{.dl-indented}\n1. domáca úloha\n: :::div{.dl-indented}\n :icon[file-earmark-ruled-fill]{.text-primary} Pracovný hárok\n : du01 :small[(odkaz otvorí hárok v správnej vetve)]\n\n :icon[deadline] Termín odovzdania\n : nedeľa 1. 3. 2026 o 23:59:59\n :::\n::::", "comments": [] }, { "id": 12, "type": "text", "data": "## Cvičenie 2.:cnt[a]\n:small[:icon[source]\nZbierka:\n:xref[2.2.1]{.exbook label="príklad"},\n:xref[2.2.2]{.exbook label="úloha"}.\nPrednáška:\n:xref[2.21]{.lec label="def."},\npríklady :xref[2.22]{.lec},\n:xref[2.23]{.lec},\n:xref[2.24]{.lec}.]\n\nMajme zadanú štruktúru $\Struct=(D, i)$, kde\n\n$\begin{aligned}\n D &= \{ 1,2,3,4,5,6 \},\n \\\n i(\sym{Alex}) &= 1, \quad\n i(\sym{Bruno}) = 2, \quad\n i(\sym{Hugo}) = 5, \quad\n i(\sym{Tina}) = 6, \quad\n \\\n i(\sym{woman}) &= \{ 1, 3, 4, 6 \},\n \\\n i(\sym{man}) &= \{ 2, 4 \},\n \\\n i(\sym{likes}) &= \{ (1,1), (1,2), (1,5), (1,6), (2,2), (3,3), (3,4), (4,4), (5,5), (5,6)\},\n \\\n i(\sym{brother}) &= \{ (1,2), (2,1), (3,1), (4,4), (5,6),(6,1), (6,2), (6,6) \},\n \\\n i(\sym{parent}) &= \{ (1,1), (2,5), (2,6), (1,5), (3,4), (4,2), (1,6), (5,6), (6,5) \},\n \\\n i(\sym{older}) &= \{ (2,1), (5,6), (6,5) \}.\n\end{aligned}$", "comments": [] }, { "id": 38, "type": "text", "data": "::h3[a)]{.float-start.h6.lh-base.my-0.me-2}\nZistite postupom zdola nahor, \nči sú formuly $A_1$ a $A_2$ pravdivé v štruktúre $\Struct$:\n\n$A_1 = (\sym{older}(\sym{Bruno},\sym{Alex}) \limpl \neg \sym{older}(\sym{Alex},\sym{Bruno}))$\n\n$A_2 = (\neg \sym{likes}(\sym{Alex},\sym{Bruno}) \lequiv \neg \sym{likes}(\sym{Bruno},\sym{Alex}))$\n\n::::aside{.note.bg-transparent.p-0} \n:icon[explanation] Svoj postup zdola nahor môžete reprezentovať tabuľkou\n\n\n| | ... | ... | ... | (older(Bruno,Alex) → ¬older(Alex,Bruno)) |\n|---|------|------|-----|------------------------------------------|\n| M | ⊨/⊭ | ⊨/⊭ | ... | ⊨/⊭ |\n\n\nv ktorej záhlaví je vytvárajúca postupnosť $A_{i,1}\,$, $A_{i,2}\,$, … formuly $A_i$.\n\n:icon[help]\nPre formulu $A_1$ sme tabuľku pripravili.\n\n:icon[warning]\nPre formulu $A_2$ potrebujete tabuľku s 8 stĺpcami,\npretože $(A \lequiv B)$ nie je spojka,\nale skratka za $((A \limpl B) \land (B \limpl A))$.\n\n\n:icon[technical]\nTrocha krajšiu tabuľku vám pomôže v značkovacom jazyku Markdown naformátovať napríklad\nMarkdown Table Editor and Generator.\n::::", "comments": [] }, { "id": 6, "type": "text", "data": ":icon[write] Riešenie:\n\n$A_1:$ \nNech $oAB = {\sf older(Alex,Bruno)}$ a $oBA = {\sf older(Bruno,Alex)}$.\n:::div{.small}\n\n| | oBA | oAB | ¬oAB | (oBA → ¬oAB) |\n|---|-----|-----|------|--------------|\n| M | ⊨/⊭ | ⊨/⊭ | ⊨/⊭ | ⊨/⊭ |\n\n:::\n\n$A_2:$\nNech $lAB = {\sf likes(Alex,Bruno)}$ a $lBA = {\sf likes(Bruno,Alex)}$.\n:::div{.small}\n\n| | lAB | ... | ... | ... | ... | ... | (¬lAB ↔ ¬lBA) |\n|---|-----|-----|-----|-----|-----|-----|---------------|\n| M | ⊨/⊭ | ⊨/⊭ | ⊨/⊭ | ⊨/⊭ | ⊨/⊭ | ⊨/⊭ | ⊨/⊭ |\n\n:::", "comments": [] }, { "id": 13, "type": "text", "data": "::h3[:icon[tools] b)]{.float-start.h6.lh-base.my-0.me-2}\nTipnite si, či je formula formula $A_3$ pravdivá v štruktúre $\Struct$:\n\n$A_3 = \bigl(\n (\sym{parent}(\sym{Bruno},\sym{Hugo}) \land\n \sym{parent}(\sym{Bruno},\sym{Tina})) \limpl\\\qquad\quad\n \bigl(\n (\neg \sym{woman}(\sym{Hugo}) \land \sym{man}(\sym{Hugo})) \limpl\n \sym{brother}(\sym{Hugo},\sym{Tina})\n \bigr)\n \bigr)$\n\nOverte svoj tip pomocou Henkinovej-Hintikkovej hry\nv prieskumníku štruktúr.\nAk ste si tipli správne,\nzahrajte si hru druhýkrát s nesprávnym tipom.\n\n:icon[homework]\nSkontrolujte pomocou Henkinovej-Hintikkovej hry,\nči ste správne určili pravdivosť formúl $A_1$ a $A_2$.\n\n:::aside{.note.bg-transparent.p-0}\n:icon[explanation] Vysvetlenie.\nHenkinova-Hintikkova hra vás vedie vyhodnotením formuly\npostupom zhora nadol\n(:xref[2.24]{.lec label="príklad"} z prednášky).\nČítajte správy a uvedomujte si,\nako postup súvisí s :xref[2.21]{.lec label="definíciou"}.\n\n* Vyhráte, ak si na začiatku tipnete správne\n a správne odpoviete aj na všetky otázky o pravdivosti podformúl,\n ktoré vám protihráč položí.\n* Prehráte, ak je váš úvodný tip nesprávny alebo si počas hry vyberiete\n čo i len jednu nesprávnu možnosť.\n Protihráč vtedy nájde cestu vytvárajúcim stromom formuly,\n ktorou vás dovedie k atómu,\n ktorý by podľa vašich odpovedí bol v štruktúre $\Struct$\n opačne pravdivý ako v skutočnosti.\n\n:icon[technical] Návod.\nFormulu pridajte do časti Truth of formulas in 𝓜 v prieskumníku tlačidlom :span[:icon[plus-lg] Add]{.p-1.rounded.border.border-success.text-success.text-nowrap}.\nSvoj tip o pravdivosti formuly vyberte z menu :span[⊨/⊭?:icon[chevron-down]{.ms-1}]{.p-1.rounded.border.text-secondary.border-secondary.text-nowrap}.\nOtvorte Henkinovu-Hintikkovu hru tlačidlom\n:span[Verify]{.p-1.rounded.border.text-danger.border-danger}.\nKeď prehráte, môžete buď zmeniť tip o pravdivosti formuly alebo niektorú zo svojich odpovedí, vedľa ktorej je odkaz :span[Change]{.text-primary.text-decoration-underline}.\n\n:icon[explanation] Vyskúšajte zobrazenie interpetácie binárnych predikátov vo forme orientovaných grafov: :span[:icon[pencil-fill] :small[:icon[caret-down-fill]{.small}]]{.p-1.rounded.border.text-nowrap} > Graphs > Oriented.\n::::", "comments": [] }, { "id": 39, "type": "newStructureExplorer", "data": { "version": 1, "formulas": { "allFormulas": [] }, "language": { "constants": { "value": [ "Alex", "Bruno", "Hugo", "Tina" ], "locked": true }, "predicates": { "value": [ [ "woman", 1 ], [ "man", 1 ], [ "likes", 2 ], [ "brother", 2 ], [ "parent", 2 ], [ "older", 2 ] ], "locked": true }, "functions": { "value": [], "locked": true }, "editMode": false }, "variables": { "value": [], "locked": true }, "teacherMode": { "teacherMode": false }, "structure": { "domain": { "value": [ "1", "2", "3", "4", "5", "6" ], "locked": true }, "iC": { "Alex": { "value": "1", "locked": true }, "Bruno": { "value": "2", "locked": true }, "Hugo": { "value": "5", "locked": true }, "Tina": { "value": "6", "locked": true } }, "iP": { "woman": { "value": [ [ "1" ], [ "3" ], [ "4" ], [ "6" ] ], "locked": true }, "man": { "value": [ [ "2" ], [ "4" ] ], "locked": true }, "likes": { "value": [ [ "1", "1" ], [ "1", "2" ], [ "1", "5" ], [ "1", "6" ], [ "2", "2" ], [ "3", "3" ], [ "3", "4" ], [ "4", "4" ], [ "5", "5" ], [ "5", "6" ] ], "locked": true }, "brother": { "value": [ [ "1", "2" ], [ "2", "1" ], [ "3", "1" ], [ "4", "4" ], [ "5", "6" ], [ "6", "1" ], [ "6", "2" ], [ "6", "6" ] ], "locked": true }, "parent": { "value": [ [ "1", "1" ], [ "2", "5" ], [ "2", "6" ], [ "1", "5" ], [ "3", "4" ], [ "4", "2" ], [ "1", "6" ], [ "5", "6" ], [ "6", "5" ] ], "locked": true }, "older": { "value": [ [ "2", "1" ], [ "5", "6" ], [ "6", "5" ] ], "locked": true } }, "iF": {} }, "graphView": { "predicate-likes": { "oriented": { "1": [ 253, 134 ], "2": [ 50, 73 ], "3": [ 194, 559 ], "4": [ 50, 404 ], "5": [ 447, 50 ], "6": [ 424, 259 ] } }, "predicate-brother": { "oriented": { "1": [ 80, 252 ], "2": [ 50, 50 ], "3": [ 112, 451 ], "4": [ 563, 50 ], "5": [ 373, 360 ], "6": [ 265, 159 ] } }, "predicate-parent": { "oriented": { "1": [ 170, 215 ], "2": [ 471, 83 ], "3": [ 50, 304 ], "4": [ 271, 320 ], "5": [ 278, 49 ], "6": [ 387, 236 ] } }, "predicate-older": { "oriented": { "1": [ 259, 80 ], "2": [ 50, 50 ], "3": [ 384, 225 ], "4": [ 50, 525 ], "5": [ 194, 380 ], "6": [ 50, 225 ] } } }, "editorToolbar": { "predicate-likes": { "openedEditor": "text", "selectedUnary": [], "unaryFilterDomain": false }, "predicate-brother": { "openedEditor": "text", "selectedUnary": [], "unaryFilterDomain": false }, "predicate-parent": { "openedEditor": "text", "selectedUnary": [], "unaryFilterDomain": false }, "predicate-older": { "openedEditor": "text", "selectedUnary": [], "unaryFilterDomain": false } } }, "comments": [] }, { "id": 2, "type": "text", "data": "## Cvičenie 2.:cnt[a]\n:small[:icon[source]\nZbierka:\n:xref[2.2.3]{.exbook label="príklad"},\n:xref[2.2.4]{.exbook label="úloha"}.]\n\n:icon[tools] Pomocou prieskumníka štruktúr vytvorte štruktúru, ktorá je modelom teórie $T=\{A_1,\dots,A_6\}$, kde\n\n$A_1 = \sym{title}(\sym{Sophie's\_Choice}) % je ok mat apostrof v nazve konstanty?$\n\n$A_2 = \sym{book}(\sym{k325})$\n\n$A_3 = \sym{has\_author}(\sym{Sophie's\_Choice,Styron})$\n\n$A_4 = (\sym{title}(\sym{The\_Catcher\_in\_the\_Rye})\land\sym{has\_author}(\sym{The\_Catcher\_in\_the\_Rye},\sym{Salinger}))% preco je tam automaticky line break??$\n\n$\begin{aligned}\n A_5 = \bigl(\n \lnot&\bigl(\sym{reads}(\sym{Adam}, \sym{k325}) \land\n \sym{admires}(\sym{Dana}, \sym{Adam})\bigr)\n \limpl \\\n &\lnot\bigl(\sym{has\_title}(\sym{k325}, \sym{Sophie's\_Choice}) \lor\n \sym{has\title}(\sym{k325}, \sym{The\Catcher\in\the\Rye})\bigr)\n \bigr)\n \end{aligned}$\n\n$A_6 = (\sym{has\title}(\sym{k325}, \sym{The\Catcher\in\the\Rye}) \lequiv \neg\sym{has\title}(\sym{k325},\sym{Sophie's\Choice}))$ \n\n:::aside{.note.bg-transparent.p-0}\n:icon[help]\nAby ste zistili, ako majú byť v štruktúre interpretované predikáty,\nanalyzujte význam formúl podľa :xref[2.21]{.lec label="definície pravdivosti"}\npostupom zhora nadol (:xref[2.24]{.lec label="príklad"} z prednášky).\nPomôže vám v tom Henkinova-Hintikkova hra, ktorú otvoríte tlačidlom\n:span[Verify]{.p-1.rounded.border.text-danger.border-danger} pod formulou.\nNajprv však musíte zadefinovať doménu štruktúry a interpretovať konštanty.\n:::", "comments": [] }, { "id": 42, "type": "newStructureExplorer", "data": { "version": 1, "formulas": { "allFormulas": [ { "text": "title(Sophies_Choice)", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "book(k325)", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "has_author(Sophies_Choice,Styron)", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "(title(The_Catcher_in_the_Rye)∧has_author(The_Catcher_in_the_Rye,Salinger))", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "(¬(reads(Adam,k325)∧admires(Dana,Adam))→¬(has_title(k325,Sophies_Choice)∨has_title(k325,The_Catcher_in_the_Rye)))", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "(has_title(k325,The_Catcher_in_the_Rye)<->¬has_title(k325,Sophies_Choice))", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] } ] }, "language": { "constants": { "value": [ "Sophies_Choice", "k325", "Styron", "The_Catcher_in_the_Rye", "Salinger", "Adam", "Dana" ], "locked": true }, "predicates": { "value": [ [ "title", 1 ], [ "book", 1 ], [ "has_author", 2 ], [ "reads", 2 ], [ "admires", 2 ], [ "has_title", 2 ] ], "locked": true }, "functions": { "value": [], "locked": true }, "editMode": false }, "variables": { "value": [], "locked": false }, "teacherMode": { "teacherMode": false }, "structure": { "domain": { "value": [], "locked": false }, "iC": {}, "iP": {}, "iF": {} }, "graphView": {}, "editorToolbar": {} }, "comments": [] }, { "id": 24, "type": "text", "data": "## Cvičenie 2.:cnt[a]\n\n:small[:icon[source]\nZbierka:\npríklady :xref[2.4.1]{.exbook},\n:xref[2.4.2]{.exbook},\n:xref[2.4.3]{.exbook label="úloha"}.\nPrednášky: \n:xref[2.35]{.lec label="def."}, \ntvrdenia \n:xref[2.36]{.lec}, :xref[2.38]{.lec},\n:xref[2.37]{.lec label=príklad}.]\n\n::h3[a)]{.float-start.h6.lh-base.my-0.me-2}\nNech $\Lang$ je jazyk výrokovologickej časti logiky prvého rádu, kde \n\n$\Consts = \{\sym{Jack}$, $\sym{Corona}\}$,\\n$\Preds = \{\asym{1}{beer}$, $\asym{2}{drink}\}$.\n \nNech $\Struct = (D,i)$ je štruktúra pre jazyk $\Lang$, kde:\n\n$$\n\begin{align*} \nD &= \{ s1, s2, s3, p1, p2 \}\n\\\ni(\sym{Jack}) &= s3,\n \\\n i(\sym{Corona}) &= p1,\n \\\n i(\sym{beer}) &= \{ p1, p2 \},\n \\\n i(\sym{drink}) &= \{ (s1,p1), (s2,p1), (s2,p2) \}\n\end{align*}\n$$\n\nZostrojte výrokovologické ohodnotenie $v$ pre $\Lang$ zhodné so štruktúrou $\Struct$.", "comments": [] }, { "id": 27, "type": "text", "data": ":icon[write]\nRiešenie:\n\n$$v = \{ ... \mapsto ..., ... \} $$", "comments": [] }, { "id": 25, "type": "text", "data": "::h3[b)]{.float-start.h6.lh-base.my-0.me-2}\nNech $\Lang$ je jazyk výrokovologickej časti logiky prvého rádu, kde:\n > $\Consts = \{\sym{Andy}$, $\sym{Woody}\}$,\\n > $\Preds = \{\asym{1}{toy}$, $\asym{1}{boy}$, $\asym{2}{plays\with}\}$.\n\n Nech\n$$\n\begin{align*} \n v = \{ &\sym{toy}(\sym{Woody}) \mapsto t,&\n &\sym{toy}(\sym{Andy}) \mapsto f,\\\n &\sym{boy}(\sym{Andy}) \mapsto t,&\n &\sym{boy}(\sym{Woody}) \mapsto f,\\\n &\sym{plays\with}(\sym{Andy},\sym{Woody}) \mapsto t,&\n &\sym{plays\with}(\sym{Woody},\sym{Andy}) \mapsto f\}\n\end{align*}\n$$\nje čiastočné ohodnotenie predikátových atómov jazyka $\Lang$.\n \nZostrojte štruktúru $\Struct$ zhodnú s $v$ na $\dom v$\n(t.j., definičnom obore ohodnotenia $v$).\nPoužite postup z prednášky (:xref[2.38]{.lec label=tvrdenie},\n:xref[2.37]{.lec label=príklad}).\n\n:icon[tools]\nSvoje riešenie overte pomocou aplikácie Structure explorer.\nJe nastavená tak, aby skontrolovala zhodu štruktúry s ohodnotením na $\dom v$.\n", "comments": [] }, { "id": 43, "type": "newStructureExplorer", "data": { "version": 1, "formulas": { "allFormulas": [ { "text": "toy(Woody)", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "boy(Andy)", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "plays_with(Andy,Woody)", "guess": true, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "toy(Andy)", "guess": false, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "boy(Woody)", "guess": false, "locked": true, "lockedGuess": true, "gameChoices": [] }, { "text": "plays_with(Woody,Andy)", "guess": false, "locked": true, "lockedGuess": true, "gameChoices": [] } ] }, "language": { "constants": { "value": [ "Andy", "Woody" ], "locked": true }, "predicates": { "value": [ [ "toy", 1 ], [ "boy", 1 ], [ "plays_with", 2 ] ], "locked": true }, "functions": { "value": [], "locked": true }, "editMode": false }, "variables": { "value": [], "locked": false }, "teacherMode": { "teacherMode": false }, "structure": { "domain": { "value": [], "locked": false }, "iC": {}, "iP": {}, "iF": {} }, "graphView": {}, "editorToolbar": {} }, "comments": [] }, { "id": 4, "type": "text", "data": "## Cvičenie 2.:cnt[a]\n:small[:icon[source]\nZbierka: :xref[2.3.1]{.exbook label="úloha"}.]\n\n:icon[tools] Pomocou aplikácie Formalization Checker sformalizujte nižšie uvedené tvrdenia v uvedenom jazyku výrokovologickej časti logiky prvého rádu $\Lang$.\n\nZamýšľaný význam predikátových symbolov je:\n\n:::div{.table.table-intended-meaning.small}\n| Predikátový symbol | Zamýšľaný význam |\n|--------------------------|---------------------------------------------|\n| works($x$) | $x$ pracuje |\n| student($x$) | $x$ je študent\*ka |\n| professor($x$) | $x$ je profesor\*ka |\n| has_classmate($x$, $y$) | $x$ má spolužiaka\*čku $y$ |\n| comes_from($x$, $y$) | $x$ pochádza z $y$ |\n| supervises($x$, $y$) | $x$ je školiteľom\*kou $y$ |\n| passed($x$, $y$) | $x$ absolvoval\*a $y$ |\n| grad_exam($x$, $y$, $z$) | $x$ dostal\a na maturite z (predmetu) $y$ hodnotenie $z$ |\n:::\n\n:::aside{.note.bg-transparent.p-1}\n:icon[warning]\nDôsledne zátvorkujte všetky formuly podľa :xref[2.6]{.lec label=definície}!\nFormalization Checker akceptuje iba formuly, v ktorých každá podformula vytvorená binárnou spojkou alebo skratkou $\lequiv$ je uzavretá do príslušného páru zátvoriek.\nChybové hlásenia žiaľ nemusia byť veľmi nápomocné.\n\n:icon[help]\nRody spomínaných osôb nie sú pre formalizáciu podstatné.\n:::", "comments": [] }, { "id": 44, "type": "formalizationChecker", "data": { "exercise": { "exercise_id": 63, "title": "2026 – Cvičenie 2.4", "description": "Sformalizujte nasledujúce výroky ako formuly v nižšie opísanom jazyku výrokovologickej časti logiky prvého rádu. ", "constants": "Dominika, Fernando, Elena, Fero, Gabika, Spain, Slovakia, Databases, Programming, Logics, Informatics, Combinatorics, Slovak_l, Czech_l , 1", "predicates": "works/1 ,student/1, professor/1, has_classmate/2, comes_from/2, supervises/2, passed/2, grad_exam/3", "functions": "", "constraints": "", "parserType": "strict", "propositions": [ { "proposition_id": 2241, "proposition": "1. Študentka Dominika má spolužiaka Fernanda, ktorý je zo Španielska.", "solution": "" }, { "proposition_id": 2242, "proposition": "2. Fernando je, samozrejme, tiež študent, napriek tomu, že popri štúdiu aj pracuje.", "solution": null }, { "proposition_id": 2243, "proposition": "3. Elena je profesorka a školí Dominiku alebo Fernanda.", "solution": null }, { "proposition_id": 2244, "proposition": "4. Fernanda tiež volajú Fero.", "solution": null }, { "proposition_id": 2245, "proposition": "5. Ak Dominika absolvovala aj databázy alebo programovanie, aj logiku, tak je Elena jej školiteľkou.", "solution": null }, { "proposition_id": 2246, "proposition": "6. Pretože „byť spolužiakomčkou“ je symetrický vzťah, je Gabika Fernandovou spolužiačkou rovnako, ako je on jej spolužiakom.", "solution": null }, { "proposition_id": 2247, "proposition": "7. Fero absolvoval programovanie iba za predpokladu, že nedostal na maturite z informatiky jednotku.", "solution": null }, { "proposition_id": 2248, "proposition": "8. Ak ju nedostal, potom musel absolvovať aj kombinatoriku.", "solution": null }, { "proposition_id": 2249, "proposition": "9. Dominika určite neabsolvovala ani kurz slovenčiny, ani češtiny, pokiaľ nie je zo zahraničia.", "solution": null }, { "proposition_id": 2250, "proposition": "10. Buď Dominika alebo Gabika je zo Slovenska (no nie obe).", "solution": null } ] }, "exerciseId": 63 }, "comments": [] }, { "id": 5, "type": "text", "data": "## :icon[homework] Cvičenie 2.:cnt[a]\n:small[:icon[source]\nZbierka:\n:xref[2.1.3]{.exbook label="príklad"},\n:xref[2.1.4]{.exbook label="úloha"}.\nPrednáška:\ndefinícia :xref[2.14]{.lec}.]\n\nUvažujme formulu:\n\n$$\n\begin{aligned}\nX = \bigl(&\n (\sym{parent}(\sym{Bruno},\sym{Hugo}) \land\n \sym{parent}(\sym{Bruno},\sym{Tina}))\\\n &\limpl\bigl(\n (\neg \sym{woman}(\sym{Hugo}) \land \sym{man}(\sym{Hugo})) \limpl\n \sym{brother}(\sym{Hugo},\sym{Tina})\n \bigr)\n \bigr)\n\end{aligned}\n$$", "comments": [] }, { "id": 23, "type": "text", "data": "Zapíšte vytvárajúci strom pre formulu $X$.\n\n_Riešenie:\n\n* $X$\n * …\n * …\n\n:::aside{.note.bg-transparent.p-0.pt-3.border-top}\n:icon[explanation] Vytvárajúci strom môžete zapísať napríklad pomocou viacúrovňového \nzoznamu nasledovne:\n\n* toto je koreň stromu\n * odsadením vytvárame deti\n * viacnásobným odsadením vytvárame deti detí\n * atď.\n * toto je druhé dieťa koreňa\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\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% General\n\providecommand{\nmodels}\n {\mathrel{\mkern1.5mu{\not}\mkern-1.5mu}\models}\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}{\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", "github": { "editBranch": "tc02", "handinBranch": "" } } }