Discrete Math Quiz: Translating To Propositional Logic
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Translating To Propositional LogicQuestion 1 of 20

A university's grade appeal process states: "An appeal is automatically approved if the original grade was based on computational error or if there's evidence of bias and the student followed proper procedures, but appeals are denied if submitted after the deadline, except in cases of documented emergency or serious illness." Let AA = "appeal approved", CC = "computational error", BB = "evidence of bias", RR = "proper procedures followed", LL = "submitted after deadline", GG = "documented emergency", SS = "serious illness". What is the correct representation?

((C∨(B∧R))∧¬L)∨(G∨S)→A((C \lor (B \land R)) \land \neg L) \lor (G \lor S) \rightarrow A
A↔(C∨(B∧R))∧¬L∨(G∨S)A \leftrightarrow (C \lor (B \land R)) \land \neg L \lor (G \lor S)
(C∨(B∧R))∧¬L→A∨(G∨S)(C \lor (B \land R)) \land \neg L \rightarrow A \lor (G \lor S)
(C∨(B∧R))∧(¬L∨G∨S)→A(C \lor (B \land R)) \land (\neg L \lor G \lor S) \rightarrow A
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Discrete Math Quiz

Discrete Math Quiz: Translating To Propositional Logic

Practice Translating To Propositional Logic in Discrete Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Translating To Propositional Logic, giving you a quick way to practice the rules, question types, and explanations that matter most for Discrete Math.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A university's grade appeal process states: "An appeal is automatically approved if the original grade was based on computational error or if there's evidence of bias and the student followed proper procedures, but appeals are denied if submitted after the deadline, except in cases of documented emergency or serious illness." Let AA = "appeal approved", CC = "computational error", BB = "evidence of bias", RR = "proper procedures followed", LL = "submitted after deadline", GG = "documented emergency", SS = "serious illness". What is the correct representation?

  1. ((C∨(B∧R))∧¬L)∨(G∨S)→A((C \lor (B \land R)) \land \neg L) \lor (G \lor S) \rightarrow A
  2. A↔(C∨(B∧R))∧¬L∨(G∨S)A \leftrightarrow (C \lor (B \land R)) \land \neg L \lor (G \lor S)
  3. (C∨(B∧R))∧¬L→A∨(G∨S)(C \lor (B \land R)) \land \neg L \rightarrow A \lor (G \lor S)
  4. (C∨(B∧R))∧(¬L∨G∨S)→A(C \lor (B \land R)) \land (\neg L \lor G \lor S) \rightarrow A (correct answer)
Explanation: When translating complex conditional statements into logical expressions, you need to carefully identify all the conditions and their relationships, especially how exceptions modify the main rules. The policy has two main parts: (1) appeals are approved if there's computational error OR (bias AND proper procedures), AND (2) appeals are denied if late, EXCEPT for emergencies or illness. The key insight is that the "except" clause means late submission doesn't automatically deny an appeal when there's an emergency or serious illness. Option D correctly captures this: (C∨(B∧R))∧(¬L∨G∨S)→A(C \lor (B \land R)) \land (\neg L \lor G \lor S) \rightarrow A. The first part (C∨(B∧R))(C \lor (B \land R)) handles the approval conditions. The second part (¬L∨G∨S)(\neg L \lor G \lor S) means either the appeal wasn't late OR there's an emergency OR serious illness - this properly models that lateness is excused by emergencies. Option A incorrectly separates the emergency conditions from the lateness requirement, suggesting emergencies always guarantee approval regardless of other factors. Option B uses a biconditional (↔\leftrightarrow), which would mean these are the ONLY ways appeals get approved, but policies often have unstated discretionary factors. Option C misplaces the implication arrow, making it seem like meeting the basic conditions leads to "approval OR emergency," which doesn't make logical sense. When translating legal or policy language into logic, pay special attention to exception clauses - they typically modify existing conditions rather than create independent pathways. The word "except" usually translates to a disjunction that negates the original restriction.

Question 2

An automated trading system follows this logic: "Execute a trade when the price trend is upward and volatility is low, or when we have strong buy signals regardless of trend, but never trade during market closure even with strong signals, unless it's an emergency override situation." Let UU = upward trend, VV = low volatility, BB = strong buy signals, CC = market closed, OO = emergency override, TT = execute trade. Which expression correctly represents this trading logic?

  1. T↔(U∧V)∨B∧(¬C∨O)T \leftrightarrow (U \land V) \lor B \land (\neg C \lor O)
  2. T↔(U∧V)∨(B∧¬C)∨OT \leftrightarrow (U \land V) \lor (B \land \neg C) \lor O
  3. T↔((U∧V)∨B)∧¬C∨OT \leftrightarrow ((U \land V) \lor B) \land \neg C \lor O
  4. T↔((U∧V)∨B)∧(¬C∨O)T \leftrightarrow ((U \land V) \lor B) \land (\neg C \lor O) (correct answer)
Explanation: When translating complex logical statements into formal expressions, you need to carefully identify the main structure and how different conditions interact. This trading system has multiple conditions with specific precedence rules that require precise use of parentheses and logical operators. The trading logic has two main scenarios for executing trades: (1) upward trend AND low volatility, OR (2) strong buy signals. However, both scenarios are subject to a crucial restriction: no trading during market closure UNLESS there's an emergency override. This creates a structure where the basic trading conditions ((U∧V)∨B)((U \land V) \lor B) must be combined with the market timing constraint (¬C∨O)(\neg C \lor O) using AND logic. The correct answer D captures this perfectly: T↔((U∧V)∨B)∧(¬C∨O)T \leftrightarrow ((U \land V) \lor B) \land (\neg C \lor O). The parentheses ensure that either basic trading condition can trigger a trade, but only when the market is open OR there's an emergency override. Choice A incorrectly places the market timing constraint only on strong buy signals, not on both trading conditions. Choice B treats emergency override as a completely separate trading trigger rather than an exception to the market closure rule. Choice C makes emergency override bypass all other conditions entirely, which contradicts the "never trade during closure unless override" logic. Remember: when parsing complex logical statements, identify the main actions first, then carefully note which restrictions apply globally versus locally. Always use parentheses to group related conditions and test your expression against edge cases.

Question 3

A smart home security system operates on this principle: "The alarm triggers when there is unauthorized entry, or when motion is detected while the system is armed and no one is home, but the alarm is disabled if the master override is active, except that fire detection always triggers the alarm even when override is active." Let UU = unauthorized entry, MM = motion detected, AA = system armed, HH = someone home, RR = master override active, FF = fire detected, TT = alarm triggers. Which formula correctly represents this system?

  1. T↔(U∨(M∧A∧¬H)∨F)∧¬R∨FT \leftrightarrow (U \lor (M \land A \land \neg H) \lor F) \land \neg R \lor F
  2. T↔((U∨(M∧A∧¬H))∧¬R)∨FT \leftrightarrow ((U \lor (M \land A \land \neg H)) \land \neg R) \lor F (correct answer)
  3. T↔(U∨(M∧A∧¬H)∧¬R)∨FT \leftrightarrow (U \lor (M \land A \land \neg H) \land \neg R) \lor F
  4. T↔(U∨(M∧A∧¬H)∨F)∧(¬R∨F)T \leftrightarrow (U \lor (M \land A \land \neg H) \lor F) \land (\neg R \lor F)
Explanation: The alarm triggers for: ((unauthorized entry OR motion while armed and no one home) AND override not active) OR fire detection. Fire detection overrides the master override. Choice A has redundant fire terms. Choice C incorrectly applies the override restriction only to motion detection. Choice D uses incorrect grouping that doesn't properly handle the fire override exception.

Question 4

An AI system's decision-making protocol states: "Take action when confidence is high and risk is low, or when there is human approval regardless of confidence level, but never take action during system maintenance unless it's a critical emergency that requires immediate response, and always defer to human judgment when available." Let CC = high confidence, LL = low risk, HH = human approval, MM = system maintenance, EE = critical emergency, JJ = human judgment available, AA = take action. Which expression correctly models this decision protocol?

  1. A↔((C∧L)∨H∨E)∧¬M∧¬JA \leftrightarrow ((C \land L) \lor H \lor E) \land \neg M \land \neg J
  2. A↔((C∧L)∨H)∧(¬M∨E)∧¬JA \leftrightarrow ((C \land L) \lor H) \land (\neg M \lor E) \land \neg J
  3. A↔¬J∧((C∧L)∨H)∧(¬M∨E)A \leftrightarrow \neg J \land ((C \land L) \lor H) \land (\neg M \lor E) (correct answer)
  4. A↔(C∧L∧¬M)∨(H∧¬J)∨EA \leftrightarrow (C \land L \land \neg M) \lor (H \land \neg J) \lor E
Explanation: When translating complex logical statements into formal logic, you need to carefully parse each condition and constraint, then combine them using the appropriate logical operators. Let's break down this AI protocol step by step. The system takes action when: (1) confidence is high AND risk is low, OR (2) there's human approval (regardless of confidence). However, there are two crucial overriding conditions: (3) never act during maintenance UNLESS it's a critical emergency, and (4) always defer when human judgment is available. The correct expression is A↔¬J∧((C∧L)∨H)∧(¬M∨E)A \leftrightarrow \neg J \land ((C \land L) \lor H) \land (\neg M \lor E). This captures all conditions: ¬J\neg J ensures no action when human judgment is available, ((C∧L)∨H)((C \land L) \lor H) represents the basic action conditions, and (¬M∨E)(\neg M \lor E) means "not during maintenance OR it's an emergency." Option A incorrectly uses ¬M∧¬J\neg M \land \neg J, which would completely prohibit action during maintenance, even in emergencies. Option B places human approval HH outside the maintenance constraint, meaning the system could act on human approval even during non-emergency maintenance. Option D fundamentally misstructures the logic by making emergency EE an independent condition that ignores the human judgment constraint. The key insight is that (¬M∨E)(\neg M \lor E) elegantly captures "never during maintenance except emergencies" – it's true when either there's no maintenance OR there's an emergency. When translating complex conditional statements, identify the primary conditions first, then carefully layer in the exceptions and overrides using the appropriate logical connectives.

Question 5

A medical diagnosis protocol states: "A patient requires immediate treatment if they have severe symptoms, or if they have moderate symptoms together with risk factors, but not if they are currently stable unless their lab results are abnormal." Let VV = severe symptoms, MM = moderate symptoms, FF = has risk factors, TT = stable, LL = abnormal lab results, II = requires immediate treatment. Which translation best captures this protocol?

  1. I↔V∨(M∧F)∨(T∧L)I \leftrightarrow V \lor (M \land F) \lor (T \land L)
  2. I↔(V∨(M∧F))∧(¬T∨L)I \leftrightarrow (V \lor (M \land F)) \land (\neg T \lor L) (correct answer)
  3. I↔V∨(M∧F)∧¬T∨LI \leftrightarrow V \lor (M \land F) \land \neg T \lor L
  4. I↔(V∨(M∧F))∧¬(T∧¬L)I \leftrightarrow (V \lor (M \land F)) \land \neg(T \land \neg L)
Explanation: The protocol requires treatment when: (severe symptoms OR moderate symptoms with risk factors) AND (not stable OR abnormal labs). The stability clause "not if stable unless labs abnormal" means treatment is blocked by stability except when labs are abnormal: (¬T∨L)(\neg T \lor L). Choice A ignores the stability constraint. Choice C has incorrect operator precedence. Choice D uses double negation unnecessarily and incorrectly.

Question 6

A software license agreement states: "This software may be used for commercial purposes only if the user has purchased a commercial license AND the software is not modified in any way." Which propositional logic expression correctly represents the condition under which commercial use is permitted, where CC = commercial license purchased, MM = software is modified, and UU = commercial use permitted?

  1. U↔(C∧¬M)U \leftrightarrow (C \land \neg M) (correct answer)
  2. U→(C∧¬M)U \rightarrow (C \land \neg M)
  3. (C∧¬M)→U(C \land \neg M) \rightarrow U
  4. U↔(C∨¬M)U \leftrightarrow (C \lor \neg M)
Explanation: The statement defines exactly when commercial use is permitted: if and only if both conditions are met (commercial license AND not modified). This is a biconditional relationship: U↔(C∧¬M)U \leftrightarrow (C \land \neg M). Choice B only states that commercial use implies the conditions but not the reverse. Choice C incorrectly states that having the conditions guarantees permission. Choice D uses OR instead of AND, allowing commercial use with just a license even if modified.

Question 7

A conference registration system uses this rule: "Attendees receive a discount if they register early and are either students or members, but corporate attendees never receive discounts regardless of when they register, except corporate members who register early do get a discount." Let EE = registers early, SS = student, MM = member, CC = corporate attendee, DD = receives discount. Which formula correctly implements this rule?

  1. D↔E∧(S∨M)∧¬C∨(C∧M∧E)D \leftrightarrow E \land (S \lor M) \land \neg C \lor (C \land M \land E)
  2. D↔E∧((S∨M)∧¬C∨(C∧M))D \leftrightarrow E \land ((S \lor M) \land \neg C \lor (C \land M)) (correct answer)
  3. D↔E∧(S∨M)∧(¬C∨M)D \leftrightarrow E \land (S \lor M) \land (\neg C \lor M)
  4. D↔E∧(S∨M∨C)D \leftrightarrow E \land (S \lor M \lor C)
Explanation: Discounts require early registration AND either: (student or member) and not corporate, OR (corporate and member). The formula E∧((S∨M)∧¬C∨(C∧M))E \land ((S \lor M) \land \neg C \lor (C \land M)) captures this. Choice A incorrectly adds an extra early registration requirement for corporate members. Choice C doesn't properly separate the corporate member exception. Choice D ignores the corporate restriction entirely.

Question 8

A company's overtime policy states: "Employees are eligible for overtime pay when they work more than 40 hours per week, except that part-time employees are never eligible regardless of hours worked, and managers are only eligible if they also work weekends." Let HH = works more than 40 hours, TT = part-time employee, MM = manager, WW = works weekends, OO = eligible for overtime. Which expression correctly models this policy?

  1. O↔(H∨W)∧¬T∧¬MO \leftrightarrow (H \lor W) \land \neg T \land \neg M
  2. O↔(H∧¬T∧¬M)∨(M∧W)O \leftrightarrow (H \land \neg T \land \neg M) \lor (M \land W)
  3. O↔H∧¬T∧(M→W)O \leftrightarrow H \land \neg T \land (M \rightarrow W)
  4. O↔(H∧¬T∧(¬M∨W))O \leftrightarrow (H \land \neg T \land (\neg M \lor W)) (correct answer)
Explanation: When translating complex logical statements into symbolic logic, you need to carefully parse each condition and exception to capture the complete policy structure. The overtime policy has three key components: (1) the basic rule requires working more than 40 hours, (2) part-time employees are completely excluded, and (3) managers have an additional requirement of working weekends. Let's build this step by step. The correct answer is D: O↔(H∧¬T∧(¬M∨W))O \leftrightarrow (H \land \neg T \land (\neg M \lor W)). This captures that overtime eligibility requires working more than 40 hours AND not being part-time AND either not being a manager OR working weekends. The parenthetical (¬M∨W)(\neg M \lor W) elegantly handles the manager exception: if you're not a manager, this is automatically true; if you are a manager, you must also work weekends. Option A incorrectly uses H∨WH \lor W, suggesting you could get overtime just by working weekends without exceeding 40 hours. Option B uses ∨\lor to separate two complete conditions, implying managers only need to work weekends (not exceed 40 hours) for overtime eligibility. Option C uses the implication M→WM \rightarrow W, which would make overtime possible for managers who don't work weekends, since a false antecedent makes the implication true. The key strategy here is recognizing that exceptions within logical statements often require careful use of disjunction (∨\lor) within conjunctions (∧\land). When you see "except that" or "unless" in policy statements, look for answer choices that use nested logical operators to handle multiple conditions simultaneously.

Question 9

A research grant approval process follows this rule: "Applications are approved when they demonstrate innovation and have adequate funding requests, or when they address critical research needs regardless of innovation level, but applications from new investigators require additional peer review unless they have senior collaboration, and no application is approved without institutional support." Let II = demonstrates innovation, FF = adequate funding request, CC = addresses critical needs, NN = new investigator, PP = additional peer review completed, SS = senior collaboration, TT = institutional support, AA = application approved. Which expression correctly captures this approval process?

  1. A↔(I∧F∧T)∨(C∧T)∧(¬N∨P∨S)A \leftrightarrow (I \land F \land T) \lor (C \land T) \land (\neg N \lor P \lor S)
  2. A↔((I∧F)∨C)∧T∧(N→(P∨S))A \leftrightarrow ((I \land F) \lor C) \land T \land (N \rightarrow (P \lor S))
  3. A↔((I∧F)∨C)∧(¬N∨P∨S)∧TA \leftrightarrow ((I \land F) \lor C) \land (\neg N \lor P \lor S) \land T (correct answer)
  4. A↔((I∧F)∨C)∧(N∧(P∨S))∧TA \leftrightarrow ((I \land F) \lor C) \land (N \land (P \lor S)) \land T
Explanation: When you encounter complex logical statements in discrete math, break them into components and translate each piece systematically. This question tests your ability to convert natural language into precise logical expressions using conjunctions, disjunctions, and implications. Let's parse the approval rule step by step. Applications are approved when: (1) they have innovation AND adequate funding, OR (2) they address critical needs regardless of innovation, BUT (3) new investigators need additional peer review OR senior collaboration, AND (4) institutional support is always required. The correct expression is A↔((I∧F)∨C)∧(¬N∨P∨S)∧TA \leftrightarrow ((I \land F) \lor C) \land (\neg N \lor P \lor S) \land T. The first part (I∧F)∨C(I \land F) \lor C captures the two approval paths. The middle part ¬N∨P∨S\neg N \lor P \lor S means "either not a new investigator, or additional peer review completed, or has senior collaboration" - which correctly handles the new investigator requirement. Finally, ∧T\land T ensures institutional support is always needed. Option A incorrectly groups the conditions, making institutional support only apply to the first approval path rather than all applications. Option B uses N→(P∨S)N \rightarrow (P \lor S), which means new investigators must have peer review OR collaboration, but this doesn't account for applications from non-new investigators properly. Option D uses N∧(P∨S)N \land (P \lor S), which would require applicants to be new investigators AND have additional requirements - completely backwards. Remember: when translating "unless" statements, convert them to disjunctions using De Morgan's laws. "New investigators require X unless Y" becomes "not new investigator OR Y OR X."

Question 10

A research study concludes: "Significant results are obtained when either the sample size is large or the effect size is large, but if both are large, then additional controls are necessary for validity." Let LL = large sample size, EE = large effect size, CC = additional controls necessary, RR = significant results obtained. Which pair of statements best represents this conclusion?

  1. R↔(L∨E)R \leftrightarrow (L \lor E) and (L∧E)→C(L \land E) \rightarrow C (correct answer)
  2. R↔(L⊕E)R \leftrightarrow (L \oplus E) and (L∧E)→C(L \land E) \rightarrow C
  3. (L∨E)→R(L \lor E) \rightarrow R and (L∧E)→C(L \land E) \rightarrow C
  4. R↔(L∨E)R \leftrightarrow (L \lor E) and C→(L∧E)C \rightarrow (L \land E)
Explanation: The statement says significant results occur when either condition is met (L∨EL \lor E), establishing a biconditional with results. Additionally, when both conditions are present, controls become necessary. Choice A correctly captures both relationships. Choice B incorrectly uses XOR, excluding the case where both conditions give results. Choice C only gives one direction of the results relationship. Choice D reverses the implication for controls.

Question 11

A university scholarship committee uses this criterion: "Students qualify for the merit scholarship if they have a high GPA and either significant extracurricular involvement or demonstrated leadership, provided they are not already receiving a full-ride scholarship, but students with exceptional academic achievement qualify regardless of other factors." Let GG = high GPA, EE = significant extracurricular involvement, LL = demonstrated leadership, FF = receiving full-ride scholarship, AA = exceptional academic achievement, QQ = qualifies for merit scholarship. Which statement correctly models this criterion?

  1. Q↔G∧(E∨L)∧¬F∧AQ \leftrightarrow G \land (E \lor L) \land \neg F \land A
  2. Q↔G∧(E∨L)∧(¬F∨A)Q \leftrightarrow G \land (E \lor L) \land (\neg F \lor A)
  3. Q↔(G∧(E∨L)∨A)∧¬FQ \leftrightarrow (G \land (E \lor L) \lor A) \land \neg F
  4. Q↔(G∧(E∨L)∧¬F)∨AQ \leftrightarrow (G \land (E \lor L) \land \neg F) \lor A (correct answer)
Explanation: When translating complex logical statements into formal logic, you need to carefully identify the main structure and handle exceptions or special cases. This scholarship criterion has a primary qualification path with an important exception that bypasses normal requirements. The statement describes two ways to qualify: (1) the standard path requiring high GPA AND (extracurriculars OR leadership) AND not having a full-ride scholarship, or (2) the exception where exceptional academic achievement qualifies students regardless of other factors. The key insight is that exceptional achievement overrides all other requirements, including the full-ride restriction. Answer D correctly captures this with (G∧(E∨L)∧¬F)∨A(G \land (E \lor L) \land \neg F) \lor A. The first part represents the standard qualification path, while ∨A\lor A creates the exception that makes exceptional students eligible regardless of anything else. Answer A uses ∧A\land A instead of ∨A\lor A, which would require exceptional achievement in addition to meeting all other criteria—the opposite of what the statement says. Answer B groups the full-ride restriction with exceptional achievement as (¬F∨A)(\neg F \lor A), incorrectly suggesting that having exceptional achievement exempts you from the full-ride restriction but still requires meeting the GPA and involvement criteria. Answer C places the full-ride restriction outside the parentheses with ∧¬F\land \neg F, meaning even exceptionally achieving students would be disqualified if they have full-ride scholarships. When modeling logical statements with exceptions, look for words like "regardless" or "provided" that signal special cases, and use disjunction (OR) to represent alternative qualification paths rather than additional requirements.

Question 12

A database query returns results based on this rule: "Records are included if they have either a priority flag OR they were created after 2020, but NOT both conditions simultaneously." Let FF = record has priority flag, AA = record created after 2020, II = record is included. Which expression correctly represents this rule?

  1. I↔(F⊕A)I \leftrightarrow (F \oplus A) (correct answer)
  2. I↔(F∨A)I \leftrightarrow (F \lor A)
  3. I↔(F∧¬A)∨(¬F∧A)I \leftrightarrow (F \land \neg A) \lor (\neg F \land A)
  4. I↔¬(F∧A)I \leftrightarrow \neg(F \land A)
Explanation: The rule describes exclusive OR (XOR): records are included if they have exactly one of the two conditions, not both. This is represented by F⊕AF \oplus A or equivalently (F∧¬A)∨(¬F∧A)(F \land \neg A) \lor (\neg F \land A). Choice A uses the XOR symbol directly. Choice B allows both conditions (inclusive OR). Choice C is equivalent to A but unnecessarily complex. Choice D only excludes records with both conditions but includes records with neither.

Question 13

Consider the statement: "If a student studies hard, then they will pass the exam, unless the exam is exceptionally difficult." Let SS = student studies hard, PP = student passes exam, DD = exam is exceptionally difficult. Which translation best captures the logical structure of this statement?

  1. (S∧¬D)→P(S \land \neg D) \rightarrow P (correct answer)
  2. S→(P∨D)S \rightarrow (P \lor D)
  3. S→(P∧¬D)S \rightarrow (P \land \neg D)
  4. (S→P)∧¬D(S \rightarrow P) \land \neg D
Explanation: The statement means studying hard leads to passing, but this implication is negated when the exam is exceptionally difficult. So studying hard AND the exam not being exceptionally difficult together imply passing: (S∧¬D)→P(S \land \neg D) \rightarrow P. Choice B incorrectly suggests studying implies either passing or difficulty. Choice C suggests studying guarantees both passing and non-difficulty. Choice D incorrectly structures the logical connectives.

Question 14

A logic puzzle states: "The treasure is found if and only if you solve the riddle correctly, but solving the riddle is only possible when you have both the ancient map and the crystal key, unless you possess the master scroll which allows solving without the crystal key." Let TT = treasure found, RR = riddle solved correctly, MM = has ancient map, KK = has crystal key, SS = has master scroll. Which system of statements correctly represents this puzzle?

  1. T↔RT \leftrightarrow R and R↔(M∧K)∨SR \leftrightarrow (M \land K) \lor S
  2. T↔RT \leftrightarrow R and R↔M∧(K∨S)R \leftrightarrow M \land (K \lor S) (correct answer)
  3. T→RT \rightarrow R and R↔(M∧K)∨SR \leftrightarrow (M \land K) \lor S
  4. T↔RT \leftrightarrow R and R→(M∧K)∨SR \rightarrow (M \land K) \lor S
Explanation: The treasure is found if and only if the riddle is solved: T↔RT \leftrightarrow R. The riddle can be solved when you have the map AND (either the key OR the scroll): M∧(K∨S)M \land (K \lor S). Choice A incorrectly allows solving with just the scroll, without requiring the map. Choice C weakens the treasure condition to one-way implication. Choice D only gives one direction for riddle solving conditions.

Question 15

A cloud storage service implements this access policy: "Files can be downloaded when the user is authenticated and either owns the file or has read permission, but shared files require the owner's approval for download unless the file is public, and system administrators can always download any file regardless of ownership or permissions." Let UU = user authenticated, OO = owns file, RR = has read permission, HH = shared file, PP = owner approval, BB = file is public, DD = system administrator, WW = can download. Which formula represents this access policy?

  1. W↔(U∧(O∨R)∨D)∧(¬H∨P∨B)W \leftrightarrow (U \land (O \lor R) \lor D) \land (\neg H \lor P \lor B)
  2. W↔U∧((O∨R)∧(¬H∨P∨B)∨D)W \leftrightarrow U \land ((O \lor R) \land (\neg H \lor P \lor B) \lor D)
  3. W↔(U∧(O∨R)∧(¬H∨P∨B))∨DW \leftrightarrow (U \land (O \lor R) \land (\neg H \lor P \lor B)) \lor D (correct answer)
  4. W↔U∧(O∨R∨D)∧(¬H∨P∨B)W \leftrightarrow U \land (O \lor R \lor D) \land (\neg H \lor P \lor B)
Explanation: When translating complex access policies into logical formulas, you need to carefully identify the main conditions and their relationships. This policy has two key components: regular user access rules and an administrator override. The policy states that files can be downloaded when: (1) a user is authenticated AND either owns the file or has read permission, BUT shared files need additional approval unless public, OR (2) the person is a system administrator (who can always download). This creates two independent paths to download access. Option C correctly captures this structure: W↔(U∧(O∨R)∧(¬H∨P∨B))∨DW \leftrightarrow (U \land (O \lor R) \land (\neg H \lor P \lor B)) \lor D. The first part (U∧(O∨R)∧(¬H∨P∨B))(U \land (O \lor R) \land (\neg H \lor P \lor B)) handles regular users who must be authenticated, have ownership or read permission, AND satisfy the shared file condition (either the file isn't shared, has approval, or is public). The ∨D\lor D at the end provides the administrator override. Option A incorrectly groups the authentication and administrator conditions together with U∧(O∨R)∨DU \land (O \lor R) \lor D, which would allow unauthenticated administrators or authenticated users without proper file access. Option B has a similar grouping error, making the shared file restrictions apply to administrators. Option D incorrectly includes administrators in the ownership/permission requirement with (O∨R∨D)(O \lor R \lor D), when they should bypass these entirely. Remember that when policies have exceptions or override conditions, these typically appear as separate disjuncts (OR terms) in your formula rather than being embedded within the main conditions.

Question 16

An access control system has the following policy: "Users can access the system if they are authenticated AND (they have admin privileges OR they are accessing during business hours), provided the system is not in maintenance mode." Let AA = authenticated, PP = has admin privileges, BB = accessing during business hours, MM = system in maintenance mode, SS = access granted. Which formula correctly represents this policy?

  1. S↔(A∧(P∨B)∧¬M)S \leftrightarrow (A \land (P \lor B) \land \neg M) (correct answer)
  2. S↔(A∧P∧B∧¬M)S \leftrightarrow (A \land P \land B \land \neg M)
  3. S↔((A∧P)∨(A∧B))∧¬MS \leftrightarrow ((A \land P) \lor (A \land B)) \land \neg M
  4. S↔A∧(P∨B)∨¬MS \leftrightarrow A \land (P \lor B) \lor \neg M
Explanation: The policy requires authentication AND either admin privileges OR business hours access, AND not in maintenance mode. This translates to A∧(P∨B)∧¬MA \land (P \lor B) \land \neg M. Choice B incorrectly requires both admin privileges AND business hours. Choice C distributes incorrectly, missing the maintenance mode constraint on both terms. Choice D has incorrect precedence, making the maintenance condition independent.

Question 17

An academic policy states: "A student receives honors if they have a GPA above 3.5 and complete a thesis project, or if they have a GPA above 3.8 regardless of thesis completion, provided they have not committed any academic violations." Let HH = "receives honors", GG = "GPA above 3.5", EE = "GPA above 3.8", TT = "completes thesis", VV = "committed academic violations". Which formula correctly represents this policy?

  1. H↔((G∧T)∨E)∧¬VH \leftrightarrow ((G \land T) \lor E) \land \neg V
  2. H↔(G∧T)∨(E∧¬V)H \leftrightarrow (G \land T) \lor (E \land \neg V)
  3. ((G∧T)∨E)∧¬V→H((G \land T) \lor E) \land \neg V \rightarrow H (correct answer)
  4. H↔((G∧T)∨E)∧¬VH \leftrightarrow ((G \land T) \lor E) \land \neg V
Explanation: The statement describes sufficient conditions for receiving honors, not necessary and sufficient conditions. It tells us when a student WILL receive honors, but doesn't say these are the ONLY ways to receive honors. Therefore, we need implication (→), not biconditional (↔). The conditions are: ((GPA > 3.5 AND thesis) OR GPA > 3.8) AND no violations. Choices A and D incorrectly use biconditional. Choice B has wrong grouping of the negation.

Question 18

A smart home system follows this rule: "The lights turn on automatically if it's dark outside and someone enters the room, or if manual override is activated, but lights will not turn on if energy saving mode is active, except when manual override is used." Let LL = "lights turn on", DD = "dark outside", EE = "someone enters room", MM = "manual override activated", SS = "energy saving mode active". Which translation captures the complete logic?

  1. (D∧E)∨(M∧¬S)→L(D \land E) \lor (M \land \neg S) \rightarrow L
  2. (D∧E∧¬S)∨M→L(D \land E \land \neg S) \lor M \rightarrow L
  3. ((D∧E)∨M)∧¬S→L((D \land E) \lor M) \land \neg S \rightarrow L
  4. ((D∧E)∨M)∧(¬S∨M)→L((D \land E) \lor M) \land (\neg S \lor M) \rightarrow L (correct answer)
Explanation: When translating complex conditional statements into logical formulas, you need to carefully identify all conditions and their relationships, especially when exceptions are mentioned. Let's break down this rule systematically. The lights turn on in two main scenarios: (1) when it's dark AND someone enters, OR (2) when manual override is activated. This gives us the base condition: (D∧E)∨M(D \land E) \lor M. However, there's a crucial restriction: energy saving mode prevents lights from turning on, BUT manual override can bypass this restriction. The correct answer is D: ((D∧E)∨M)∧(¬S∨M)→L((D \land E) \lor M) \land (\neg S \lor M) \rightarrow L. The second part (¬S∨M)(\neg S \lor M) captures that either energy saving is off OR manual override is active. This ensures lights work normally when energy saving is off, but manual override can still activate lights even when energy saving is on. Answer A incorrectly suggests manual override only works when energy saving is off (M∧¬S)(M \land \neg S), missing that manual override should bypass energy saving entirely. Answer B fails to handle the energy saving restriction for the dark-and-enter condition - it would allow lights to turn on when it's dark, someone enters, AND energy saving is active (without manual override). Answer C completely blocks lights when energy saving is active, even with manual override, contradicting the "except when manual override is used" clause. Remember: when logical statements include exceptions (words like "except," "unless," "but"), look for formulations that use OR conditions to represent when those exceptions apply.

Question 19

An airline's boarding policy states: "Priority boarding is given to first-class passengers and frequent flyers with elite status, but not to passengers who booked with award miles, unless they also have elite status, and military personnel always get priority regardless of other factors." Let PP = "gets priority boarding", FF = "first-class passenger", EE = "frequent flyer with elite status", AA = "booked with award miles", II = "military personnel". Which formula correctly represents this policy?

  1. P↔(F∨E∨I)∧(¬A∨E)P \leftrightarrow (F \lor E \lor I) \land (\neg A \lor E)
  2. ((F∨E)∧(¬A∨E))∨I→P((F \lor E) \land (\neg A \lor E)) \lor I \rightarrow P (correct answer)
  3. (F∨E∨I)∧(¬A∨E)→P(F \lor E \lor I) \land (\neg A \lor E) \rightarrow P
  4. (F∨(E∧¬A))∨I→P(F \lor (E \land \neg A)) \lor I \rightarrow P
Explanation: When translating complex policy statements into logical formulas, you need to carefully parse each clause and identify the logical relationships between conditions. This policy has multiple parts that must work together. Let's break down the policy systematically. The statement gives priority boarding to: (1) first-class passengers and frequent flyers with elite status, but (2) not to award mile passengers unless they also have elite status, and (3) military personnel always get priority regardless of other factors. The correct answer is B: ((F∨E)∧(¬A∨E))∨I→P((F \lor E) \land (\neg A \lor E)) \lor I \rightarrow P. This formula captures the structure perfectly. The first part (F∨E)∧(¬A∨E)(F \lor E) \land (\neg A \lor E) handles the civilian rules: you get priority if you're first-class OR have elite status, AND you either didn't book with award miles OR you have elite status (which overrides the award mile restriction). The ∨I\lor I adds that military personnel always qualify, regardless of other factors. Option A uses a biconditional (↔\leftrightarrow), which would mean these are the ONLY ways to get priority boarding, but the policy doesn't exclude other possible reasons. Option C incorrectly applies the award mile restriction to military personnel by putting II inside the conjunction with (¬A∨E)(\neg A \lor E). Option D misrepresents the elite status exception by grouping E∧¬AE \land \neg A, suggesting elite status only helps if you didn't use award miles. When converting policies to logic, pay special attention to exception clauses like "unless" and "regardless of" – they often require careful placement of conjunctions and disjunctions to avoid unintended restrictions.

Question 20

A software system has the following requirement: "The backup process runs if and only if it's after midnight and either the system load is low or manual override is activated, but not when maintenance mode is active." Let BB = "backup process runs", MM = "after midnight", LL = "system load is low", OO = "manual override activated", NN = "maintenance mode active". What is the correct propositional logic representation?

  1. B↔(M∧(L∨O)∧¬N)B \leftrightarrow (M \land (L \lor O) \land \neg N) (correct answer)
  2. B↔(M∧(L∨O))∧¬NB \leftrightarrow (M \land (L \lor O)) \land \neg N
  3. B→(M∧(L∨O)∧¬N)B \rightarrow (M \land (L \lor O) \land \neg N)
  4. B↔((M∧L)∨(M∧O))∧¬NB \leftrightarrow ((M \land L) \lor (M \land O)) \land \neg N
Explanation: The phrase 'if and only if' indicates a biconditional (↔). The backup runs exactly when: it's after midnight AND (low load OR manual override) AND NOT in maintenance mode. Choice A correctly groups all conditions. Choice B has incorrect parentheses placement. Choice C uses implication instead of biconditional. Choice D distributes incorrectly and has wrong grouping with the negation.