gistfile1.json

"categories": [
       {
           "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": "Answer all of the following:<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": "Which of the following is more sad, and why?<ol><li>\\(\\exists x((Tx\\ \\&\\ \\forall y(Ty\\to y=x))\\ \\&\\ Lx)\\)</li><li>\\(\\exists x((Tx\\ \\&\\ Lx)\\ \\&\\ \\forall y((Ty\\ \\&\\ Ly)\\to y=x))\\)</li></ol> Assume Tx means that x is a turtle and Lx means that x is lonely.",
                   "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? Also, 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 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\\)?",
                   "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 syntax and grammar of truth-functional logic, give an example of an incomplete system.",
                   "answer": "",
                   "value": 500
               },
               {
                   "question": "Using the syntax and grammar 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
               }
           ]
       }
   ]