gistfile1.json

[
   {
       "name": "TF Logic",
       "order": 0,
       "questions": [
           {
               "question": "What does it mean for a sentence to be a tautology?",
               "answer": "Every possible interpretation makes the sentence true.",
               "value": 100
           },
           {
               "question": "Are each of the following true or false? <ol><li>A sentence is contingent iff it is falsifiable and consistent</li><li>A sentence P is a contradiction if it is not a tautology.</li><li>If a sentence is consistent, then it is falsifiable.</li></ol>",
               "answer": "T/F/F",
               "value": 200
           },
           {
               "question": "Construct a truth-table to show that \\(C\\ \\&\\ (B\\vee A)\\approx (C\\ \\&\\ B)\\vee(C\\ \\&\\ A)\\)",
               "answer": "(table)",
               "value": 300
           },
           {
               "question": "Write an explanation for why if \\(p,q\\models r\\), then the sentence \\((p\\ \\&\\ q)\\to r\\) is a tautology.",
               "answer": "(explanation)",
               "value": 400
           },
           {
               "question": "Provide a natural deduction proof to demonstrate \\(C\\to(B\\vee A),\\ \\sim B\\ \\&\\ \\sim D,\\ A\\leftrightarrow B,\\) \\(\\ ((E\\vee B)\\leftrightarrow C)\\to(A\\vee D)\\vdash E\\)",
               "answer": "(proof)",
               "value": 500
           }
       ]
   },
   {
       "name": "Q Logic",
       "order": 1,
       "questions": [
           {
               "question": "Translate the following into quantifier logic symbols: If all Cylons are evil, then Gaius Baltar is evil.",
               "answer": "",
               "value": 100
           },
           {
               "question": "Provide a natural deduction proof to show \\(\\forall x(Fx\\to Gx)\\) \\(\\vdash (\\forall x\\sim Gx\\to\\forall x\\sim Fx)\\)",
               "answer": "",
               "value": 200
           },
           {
               "question": "On a street there are two buildings. Statement 1 is true for building 1, and statement 2 is true for building 2. Which of these buildings has a master key? <ol><li>\\(\\forall x\exists y(Dx\\to(Ky\\ \\&\\ Uxy))\\)</li><li>\\(\\exists y\\forall x(Ky\\ \\&\\ (Dx\\to Uxy))\\)</li></ol>",
               "answer": "(good explanations)",
               "value": 300
           },
           {
               "question": "Consider the two sentences:<ol><li>Everybody loves somebody.</li><li>Somebody is loved by everybody.</li></ol> One implies the other. Which implies which? Prove it.",
               "answer": "",
               "value": 400
           },
           {
               "question": "Does the following interpretation show that the sentence \\(\\forall x\\exists y(Lxy\\vee Px)\\) is falsifiable? Why or why not?<ul><li>Domain: \\(\\{1,2,3\\}\\)</li><li>\\(P = \\{2,3\\}\\)</li><li>\\(L = \\{\\langle 3,3\\rangle, \\langle 1,1\\rangle, \\langle 2,1\\rangle\\}\\)</li></ul>",
               "answer": "",
               "value": 500
           }
       ]
   },
   {
       "name": "Sets",
       "order": 2,
       "questions": [
           {
               "question": "What is the definition of subset?",
               "answer": "\\(A\\subseteq B\\leftrightarrow \\forall x(x\\in A\\to x\\in B)\\)",
               "value": 100
           },
           {
               "question": "True or false: The power set of the empty set is the empty set. If true, explain why. If false, calculate the actual power set.",
               "answer": "False, \\(P(\\emptyset) = \\{\\emptyset\\}\\)",
               "value": 200
           },
           {
               "question": "What does it mean for a relation to be symmetric (by definition)? Provide an example of a symmetric relation.",
               "answer": "(defn + ex)",
               "value": 300
           },
           {
               "question": "Prove that if \\(A\\neq\\emptyset\\), then either \\(A\\setminus B\\neq\\emptyset\\) or \\(A\\cap B\\neq \\emptyset\\).",
               "answer": "(proof)",
               "value": 400
           },
           {
               "question": "Prove that if B and C are disjoint (\\(B\\cap C=\\emptyset\\)) and \\(A\\cap C\\) is nonempty, then \\(A\\not\\subseteq B\\).",
               "answer": "(proof)",
               "value": 500
           }
       ]
   },
   {
       "name": "Misc. Math",
       "order": 3,
       "questions": [
           {
               "question": "What does it mean (by definition) to say \\(3\\mid n\\)? (not 'n is divisible by 3' or '3 divides n')",
               "answer": "There is a natural number k such that 3k=n.",
               "value": 100
           },
           {
               "question": "Prove that \\(n^3\\) is even iff \\(n\\) is even.",
               "answer": "",
               "value": 200
           },
           {
               "question": "Prove, by mathematical induction, that \\(0^2 + 1^2 +\\cdots + n^2 = \\frac{n(n+1)(2n+1)}{6}\\).",
               "answer": "",
               "value": 300
           },
           {
               "question": "Prove that for any natural number \\(n\\), \\(n(n^2 - 1)\\) is divisible by 6.",
               "answer": "(proof)",
               "value": 400
           },
           {
               "question": "Prove that for any natural numbers \\(m\\) and \\(n\\), \\(mn\\) is even iff either \\(m\\) is even or \\(n\\) is even.",
               "answer": "",
               "value": 500
           }
       ]
   },
   {
       "name": "Misc. Logic",
       "order": 4,
       "questions": [
           {
               "question": "Must a valid argument have premises that are true in the real world? Why or why not?",
               "answer": "No",
               "value": 100
           },
           {
               "question": "Using the grammar and semantics of truth-functional logic, give an example of an incomplete system.",
               "answer": "",
               "value": 500
           },
           {
               "question": "Using the grammar and semantics of truth-functional logic, give an example of an unsound system.",
               "answer": "",
               "value": 400
           },
           {
               "question": "What does it mean for a logical system to be sound?",
               "answer": "",
               "value": 200
           },
           {
               "question": "What does it mean for a logical system to be complete?",
               "answer": "",
               "value": 300
           }
       ]
   }
]