// Numbas version: exam_results_page_options {"name": "Truth tables 2 ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "

Create a truth table for a logical expression of the form $((a \\operatorname{op1} b) \\operatorname{op2}(c \\operatorname{op3} d))\\operatorname{op4}e $ where each of $a, \\;b,\\;c,\\;d,\\;e$ can be one the Boolean variables $p,\\;q,\\;\\neg p,\\;\\neg q$ and each of $\\operatorname{op1},\\;\\operatorname{op2},\\;\\operatorname{op3},\\;\\operatorname{op4}$ one of $\\lor,\\;\\land,\\;\\to$.

\n

For example: $((q \\lor \\neg p) \\to (p \\land \\neg q)) \\lor \\neg q$

", "licence": "Creative Commons Attribution 4.0 International"}, "variables": {"op": {"definition": "latex(random(\"\\\\lor\",\"\\\\land\",\"\\\\to\"))", "description": "", "group": "First Bracket", "templateType": "anything", "name": "op"}, "p": {"definition": "bool_to_label([true,true,false,false])", "description": "", "group": "Truth values", "templateType": "anything", "name": "p"}, "pre_ev1": {"definition": "map(evaluate(convch(a)+\" \"+conv(op)+\" \"+convch(b),[bool_p[t],bool_q[t]]),t,0..3)", "description": "", "group": "First Bracket", "templateType": "anything", "name": "pre_ev1"}, "b": {"definition": "latex(latex_symbol_list[s[1]])", "description": "", "group": "First Bracket", "templateType": "anything", "name": "b"}, "op2": {"definition": "latex(random(\"\\\\lor\",\"\\\\land\",\"\\\\to\"))", "description": "", "group": "Second Bracket", "templateType": "anything", "name": "op2"}, "a1": {"definition": "latex(latex_symbol_list[s[2]])", "description": "", "group": "Second Bracket", "templateType": "anything", "name": "a1"}, "op1": {"definition": "latex(random(\"\\\\lor\",\"\\\\land\",\"\\\\to\"))", "description": "", "group": "First and Second Brackets", "templateType": "anything", "name": "op1"}, "q": {"definition": "bool_to_label([true,false,true,false])", "description": "", "group": "Truth values", "templateType": "anything", "name": "q"}, "logic_symbol_list": {"definition": "[\"p\",\"q\",\"not p\",\"not q\"]", "description": "", "group": "Lists of symbols", "templateType": "anything", "name": "logic_symbol_list"}, "ev1": {"definition": "bool_to_label(pre_ev1)", "description": "", "group": "First Bracket", "templateType": "anything", "name": "ev1"}, "a": {"definition": "latex(latex_symbol_list[s[0]])", "description": "", "group": "First Bracket", "templateType": "anything", "name": "a"}, "bool_p": {"definition": "[true,true,false,false]", "description": "", "group": "Truth values", "templateType": "anything", "name": "bool_p"}, "t_value": {"definition": "bool_to_label(pre_t_value)", "description": "", "group": "First and Second Brackets", "templateType": "anything", "name": "t_value"}, "pre_ev2": {"definition": "map(evaluate(convch(a1)+\" \"+conv(op2)+\" \"+convch(b1),[bool_p[t],bool_q[t]]),t,0..3)", "description": "", "group": "Second Bracket", "templateType": "anything", "name": "pre_ev2"}, "op4": {"definition": "latex(random(\"\\\\lor\",\"\\\\land\",\"\\\\to\"))", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "op4"}, "final_value": {"definition": "bool_to_label(pre_final_value)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "final_value"}, "s": {"definition": "repeat(random(0..3),6)", "description": "", "group": "Lists of symbols", "templateType": "anything", "name": "s"}, "ev3": {"definition": "bool_to_label(pre_ev3)", "description": "", "group": "Last ", "templateType": "anything", "name": "ev3"}, "ev2": {"definition": "bool_to_label(pre_ev2)", "description": "", "group": "Second Bracket", "templateType": "anything", "name": "ev2"}, "bool_q": {"definition": "[true,false,true,false]", "description": "", "group": "Truth values", "templateType": "anything", "name": "bool_q"}, "pre_final_value": {"definition": "map(evaluate(pre_t_value[t]+\" \"+conv(op4)+\" \"+pre_ev3[t],[]),t,0..3)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "pre_final_value"}, "b1": {"definition": "latex(latex_symbol_list[s[3]])", "description": "", "group": "Second Bracket", "templateType": "anything", "name": "b1"}, "pre_t_value": {"definition": "map(evaluate(pre_ev1[t]+\" \"+conv(op1)+\" \"+pre_ev2[t],[]),t,0..3)", "description": "", "group": "First and Second Brackets", "templateType": "anything", "name": "pre_t_value"}, "latex_symbol_list": {"definition": "[\"p\",\"q\",\"\\\\neg p\",\"\\\\neg q\"]", "description": "", "group": "Lists of symbols", "templateType": "anything", "name": "latex_symbol_list"}, "a2": {"definition": "latex(random(\"\\\\neg p\",\"\\\\neg q\"))", "description": "", "group": "Last ", "templateType": "anything", "name": "a2"}, "pre_ev3": {"definition": "map(evaluate(convch(a2),[bool_p[t],bool_q[t]]),t,0..3)", "description": "", "group": "Last ", "templateType": "anything", "name": "pre_ev3"}}, "advice": "

First we find the truth table for $\\var{a} \\var{op} \\var{b}$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a} \\var{op} \\var{b}$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev1[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev1[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev1[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev1[3]}$
\n

Then the truth table for $\\var{a1} \\var{op2} \\var{b1}$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a1} \\var{op2} \\var{b1}$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev2[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev2[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev2[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev2[3]}$
\n

Putting these together to find $(\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1})$:

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a} \\var{op} \\var{b}$$\\var{a1} \\var{op2} \\var{b1}$$(\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1})$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev1[0]}$$\\var{ev2[0]}$$\\var{t_value[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev1[1]}$$\\var{ev2[1]}$$\\var{t_value[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev1[2]}$$\\var{ev2[2]}$$\\var{t_value[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev1[3]}$$\\var{ev2[3]}$$\\var{t_value[3]}$
\n

Next we find the truth table for $\\var{a2}$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a2}$
$\\var{p[0]}$$\\var{q[0]}$$\\var{ev3[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{ev3[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{ev3[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{ev3[3]}$
\n

Putting this all together to obtain the truth table we want:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$(\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1})$$\\var{a2}$$((\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1}))\\var{op4}\\var{a2} $
$\\var{p[0]}$$\\var{q[0]}$$\\var{t_value[0]}$$\\var{ev3[0]}$$\\var{final_value[0]}$
$\\var{p[1]}$$\\var{q[1]}$$\\var{t_value[1]}$$\\var{ev3[1]}$$\\var{final_value[1]}$
$\\var{p[2]}$$\\var{q[2]}$$\\var{t_value[2]}$$\\var{ev3[2]}$$\\var{final_value[2]}$
$\\var{p[3]}$$\\var{q[3]}$$\\var{t_value[3]}$$\\var{ev3[3]}$$\\var{final_value[3]}$
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In the following question you are asked to construct a truth table for:

\n

\\[((\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1}))\\var{op4}\\var{a2}.\\]

\n

\n

Enter T if true, else enter F.

\n

\n

\n

\n

\n

\n

\n

\n

\n

\n

\n

", "variablesTest": {"condition": "a1 <>b1 and\nif(a='p' or a='\\\\neg p',b=random('q','\\\\neg q'),b=random('p','\\\\neg p'))\n", "maxRuns": "150"}, "parts": [{"type": "gapfill", "marks": 0, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "

Complete the following truth table:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$p$$q$$\\var{a} \\var{op} \\var{b}$$\\var{a1} \\var{op2} \\var{b1}$$(\\var{a} \\var{op} \\var{b}) \\var{op1} (\\var{a1} \\var{op2} \\var{b1})$$\\var{a2} $$((\\var{a} \\var{op} \\var{b})\\var{op1}(\\var{a1} \\var{op2} \\var{b1}))\\var{op4}\\var{a2} $
$\\var{p[0]}$$\\var{q[0]}$[[0]][[4]][[8]][[12]][[16]]
$\\var{p[1]}$$\\var{q[1]}$[[1]][[5]][[9]][[13]][[17]]
$\\var{p[2]}$$\\var{q[2]}$[[2]][[6]][[10]][[14]][[18]]
$\\var{p[3]}$$\\var{q[3]}$[[3]][[7]][[11]][[15]][[19]]
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"scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{ev2[1]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{ev2[2]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{ev2[2]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{ev2[3]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{ev2[3]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{t_value[0]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{t_value[0]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{t_value[1]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{t_value[1]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{t_value[2]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{t_value[2]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{t_value[3]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{t_value[3]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{ev3[0]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{ev3[0]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{ev3[1]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{ev3[1]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{ev3[2]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{ev3[2]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{ev3[3]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{ev3[3]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{final_value[0]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{final_value[0]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{final_value[1]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{final_value[1]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{final_value[2]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{final_value[2]}"}, {"type": "patternmatch", "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "displayAnswer": "{final_value[3]}", "matchMode": "regex", "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{final_value[3]}"}], "sortAnswers": false, "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true}], "extensions": [], "functions": {"evaluate": {"type": "number", "definition": "return scope.evaluate(expr);", "parameters": [["expr", "string"], ["dependencies", "list"]], "language": "javascript"}, "bool_to_label": {"type": "number", "definition": "map(if(l[x],'T','F'),x,0..length(l)-1)", "parameters": [["l", "list"]], "language": "jme"}, "convch": {"type": "string", "definition": "switch(ch=\"\\\\neg p\",\"not bool_p[t]\",ch=\"\\\\neg q\",\"not bool_q[t]\",ch=\"p\",\"bool_p[t]\",\"bool_q[t]\")", "parameters": [["ch", "string"]], "language": "jme"}, "conv": {"type": "string", "definition": "switch(op=\"\\\\land\",\"and\",op=\"\\\\lor\",\"or\",\"implies\")", "parameters": [["op", "string"]], "language": "jme"}}, "type": "question", "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}]}]}], "contributors": [{"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}]}