Inference and Deduction

215 soru

Soru 21Soru →

In an effort to reduce urban noise pollution, the city of Oakhaven replaced all of its traditional diesel buses with electric buses. Electric buses produce significantly less engine noise than diesel buses. However, electric buses are heavier and cause more tire-to-road friction noise at speeds above 40 km/h than diesel buses do. Most of the bus routes in Oakhaven's residential neighborhoods have speed limits of 30 km/h and are designed with frequent stops, preventing buses from exceeding these limits. Conversely, the bus routes connecting Oakhaven's commercial districts to the suburbs often require buses to travel at speeds up to 60 km/h for extended stretches.

Which one of the following is most strongly supported by the information above?

Cevabı ve açıklamayı göster

Cevap: The replacement of diesel buses with electric buses likely resulted in a greater reduction in bus-related noise in Oakhaven's residential neighborhoods than on the routes connecting commercial districts to the suburbs.

Cevap

The statement regarding the greater reduction in bus-related noise in residential neighborhoods compared to suburban routes is the most strongly supported.
The correct answer accurately synthesizes the provided facts. Because residential routes prevent buses from exceeding 30 km/h, electric buses there do not trigger the increased friction noise that occurs above 40 km/h, meaning these areas experience the full benefit of reduced engine noise. In contrast, suburban routes require speeds up to 60 km/h, where the electric buses' increased friction noise will offset some or all of the reduced engine noise. Therefore, it is highly supported that the residential areas experienced a greater net reduction in bus-related noise.

Adım Adım Çözüm

1
Analyze the noise profiles of the two bus types based on the text.
Electric buses have less engine noise than diesel buses, but they have more friction noise when traveling faster than 40 km/h.
Understanding the precise conditions under which each bus type is noisier is necessary to draw valid conclusions.
2
Analyze the conditions of the different routes.
Residential routes keep buses under 40 km/h (speed limit 30 km/h). Suburban routes require buses to travel up to 60 km/h.
The different routes trigger different aspects of the buses' noise profiles.
3
Synthesize the noise profiles with the route conditions.
In residential areas, electric buses are quieter (less engine noise, no excess friction noise). In suburban areas, they produce excess friction noise, mitigating the benefit of less engine noise.
This synthesis directly proves that the net reduction in noise is greater in residential areas.

Anahtar Kavram

Synthesizing conditional facts to derive a comparative conclusion
Soru 22Soru →

In ancient Mediterranean societies, the widespread adoption of standardized silver coinage often occurred concurrently with an increase in long-distance maritime trade. However, historians note that the earliest standardized coins were minted almost exclusively in small denominations, which were highly impractical for large-scale maritime transactions but well-suited for daily wage labor and local retail markets. Furthermore, long-distance trade had flourished for centuries prior to the introduction of coinage, relying on weighed bullion and credit systems.

Which of the following statements is most strongly supported by the information above?

Cevabı ve açıklamayı göster

Cevap: During the initial period when standardized silver coinage was widely adopted, merchants engaged in large-scale maritime transactions likely relied on methods of exchange other than the newly minted coins.

Cevap

During the initial period when standardized silver coinage was widely adopted, merchants engaged in large-scale maritime transactions likely relied on methods of exchange other than the newly minted coins.
The stimulus states that the earliest standardized coins were 'highly impractical for large-scale maritime transactions' and that maritime trade had already flourished using 'weighed bullion and credit systems'. By combining these premises, it is strongly supported that merchants conducting large-scale maritime transactions during this initial period likely used methods other than the new, impractical small-denomination coins.

Adım Adım Çözüm

1
Analyze the characteristics of the earliest standardized coins.
The earliest standardized coins were minted almost exclusively in small denominations, making them highly impractical for large-scale maritime transactions.
To determine whether the newly introduced coins could realistically be used for the expanding long-distance trade.
2
Evaluate how long-distance maritime trade operated at the time.
Long-distance trade had already flourished for centuries before coinage by utilizing weighed bullion and credit systems.
To identify if alternative exchange methods existed and were accessible to maritime merchants.
3
Synthesize the findings to derive a logical inference.
Because the early standardized coins were impractical for large-scale trade, and established non-coin methods already existed, it is highly likely that maritime merchants continued using these other methods during the initial adoption of coinage.
To identify the statement most strongly supported by combining all the provided premises.

Anahtar Kavram

Most Strongly Supported Inferences
Soru 23Soru →

In the city of Oakhaven, the municipal recycling facility processes only plastics labeled type 1 and type 2. If a collected batch of recyclable materials contains any other types of plastic, the facility must either employ manual sorters to remove the unaccepted items or send the entire batch to the landfill. Due to severe municipal budget cuts this year, the facility currently has no manual sorters and has no means to acquire them. Nevertheless, recent audits indicate that more than half of the recycling batches collected from Oakhaven's residential bins contain plastics other than types 1 and 2.

Which one of the following statements is most strongly supported by the information above?

Cevabı ve açıklamayı göster

Cevap: More than half of the recycling batches collected from Oakhaven's residential bins will be sent to the landfill.

Cevap

More than half of the recycling batches collected from Oakhaven's residential bins will be sent to the landfill.
The correct answer synthesizes the facts perfectly. The stimulus establishes that if a batch contains unacceptable plastics, it must either be manually sorted or sent to the landfill. We are explicitly told that manual sorting is impossible because the facility has no sorters and no means to acquire them. Therefore, any batch with unacceptable plastics must be sent to the landfill. Since the audits show that more than half of the residential batches contain unacceptable plastics, it logically follows that more than half of these batches must be sent to the landfill.

Adım Adım Çözüm

1
Identify the conditional rule regarding contaminated batches.
If a batch has plastics other than type 1 and 2, it must be manually sorted OR sent to the landfill.
This establishes the logical consequences for batches containing unaccepted materials.
2
Apply the factual constraints provided in the stimulus.
The facility has no manual sorters and no means to acquire them.
This eliminates the 'manual sorting' option, leaving 'sent to the landfill' as the only possible outcome for contaminated batches.
3
Combine the rule with the quantitative data provided by the audits.
Since more than half of the residential batches are contaminated, and all contaminated batches must go to the landfill, more than half of the batches must go to the landfill.
This yields a deduction that must be true based entirely on the provided premises.

Anahtar Kavram

Synthesizing conditional statements with quantitative premises to form a valid deduction.
Soru 24Soru →

Linguists studying the Voro language family have observed that Voro languages that evolved in coastal environments possess a highly specialized vocabulary for distinct types of ocean currents. In contrast, Voro languages that evolved in inland, arid regions almost entirely lack such vocabulary. The modern language of Zani, which belongs to the Voro family and is spoken exclusively by a nomadic population in a landlocked desert, nonetheless contains an extensive array of archaic terms for ocean currents. Historical texts provide strong evidence that the Zani people have never traded or interacted with coastal populations.

Which one of the following is most strongly supported by the information above?

Cevabı ve açıklamayı göster

Cevap: The Zani language likely underwent at least some of its development in a coastal region.

Cevap

The statement that the Zani language likely underwent at least some of its development in a coastal region is the most strongly supported inference.
The stimulus establishes a mystery: Zani possesses a specialized coastal vocabulary despite being an inland language, and it explicitly eliminates trade as a source for this vocabulary. Since inland Voro languages generally lack such terms, the most logical explanation strictly supported by the text is that Zani acquired this vocabulary by developing, at least partially, in a coastal region before its speakers became completely landlocked.

Adım Adım Çözüm

1
Analyze the premises provided in the stimulus.
Coastal Voro languages have ocean current vocabulary; inland Voro languages generally lack it. Zani is a landlocked Voro language with this vocabulary. Zani people never interacted with coastal populations.
To establish the factual constraints and relationships of the argument.
2
Evaluate the anomaly presented.
Zani has coastal vocabulary despite being an inland language whose speakers have no history of coastal trade.
To identify the logical gap that the correct inference must resolve based solely on the text.
3
Synthesize the premises to find the most strongly supported conclusion.
Since Zani could not have acquired the vocabulary through trade, and inland Voro languages don't typically develop it, the language must have acquired it by developing in a coastal region before its speakers became fully landlocked.
To derive a logical inference that explains all the provided facts without relying on outside assumptions.

Anahtar Kavram

Most Strongly Supported Questions
Soru 25Soru →

Any infrastructure project that utilizes synthetic materials must pass a secondary safety inspection. Furthermore, no project that passes a secondary safety inspection is completed within its original budget. Every project managed by the transit authority utilizes synthetic materials, and some projects managed by the transit authority are funded by state bonds. Finally, only projects that incorporate sustainable design are funded by state bonds.

If the statements above are true, which one of the following must also be true?

Cevabı ve açıklamayı göster

Cevap: At least one project that incorporates sustainable design fails to be completed within its original budget.

Cevap

The correct answer states that at least one project that incorporates sustainable design fails to be completed within its original budget.
The correct option is provable by connecting multiple formal premises. We know that some transit authority projects are funded by state bonds, and because ONLY projects with sustainable design receive state bonds, those specific transit projects must incorporate sustainable design. Furthermore, every transit project utilizes synthetic materials and thus passes a secondary safety inspection, which guarantees they are not completed within budget. Therefore, there exists at least one project (the state-bond-funded transit project) that incorporates sustainable design and is not completed within its original budget.

Adım Adım Çözüm

1
Map the universal conditional chain starting from the transit authority.
Transit Authority -> Synthetic Materials -> Secondary Safety Inspection -> NOT Completed Within Budget.
This establishes that every single project managed by the transit authority ultimately fails to be completed within its original budget.
2
Translate the 'only' statement into a standard conditional.
The phrase 'only projects that incorporate sustainable design are funded by state bonds' translates to: State Bonds -> Sustainable Design.
The word 'only' modifies the necessary condition, meaning sustainable design is required for state bond funding.
3
Link the 'some' statement to the established conditions.
We are told: Transit Authority <-some-> State Bonds. Using the translation from Step 2, this means: Transit Authority <-some-> Sustainable Design.
Because some transit projects are funded by state bonds, and all state bond projects have sustainable design, at least some transit projects must have sustainable design.
4
Synthesize the overlapping groups to find the final necessary truth.
From Step 3, there is an intersecting group of projects that are both Transit Authority AND Sustainable Design. From Step 1, ALL Transit Authority projects go over budget. Therefore, this intersecting group of Sustainable Design projects must also go over budget.
Any characteristic that applies to ALL transit authority projects must also apply to the subset of transit authority projects that incorporate sustainable design.

Anahtar Kavram

Must Be True Questions
Soru 26Soru →

According to a corporation's restructuring plan, if the board of directors approves the merger, then the marketing division will be downsized. Any downsizing of the marketing division requires that the public relations team be outsourced. Furthermore, the company will undergo a comprehensive financial audit only if the public relations team is not outsourced.

Based on the corporation's restructuring plan, which one of the following must be true?

Cevabı ve açıklamayı göster

Cevap: If the board of directors approves the merger, the company will not undergo a comprehensive financial audit.

Cevap

If the board of directors approves the merger, the company will not undergo a comprehensive financial audit.
The stimulus provides three conditional statements: (1) Merger Approved -> Marketing Downsized, (2) Marketing Downsized -> PR Outsourced, and (3) Financial Audit -> PR NOT Outsourced. By taking the contrapositive of the third statement, we get: PR Outsourced -> NO Financial Audit. Linking all these together creates a continuous chain: Merger Approved -> Marketing Downsized -> PR Outsourced -> NO Financial Audit. Therefore, if the merger is approved, the company absolutely cannot undergo a financial audit.

Adım Adım Çözüm

1
Translate the first two premises into formal conditional logic.
Merger Approved -> Marketing Downsized; Marketing Downsized -> PR Outsourced.
To establish the initial chain of sufficient and necessary conditions.
2
Translate the third premise into formal conditional logic and find its contrapositive.
Audit -> PR NOT Outsourced. The contrapositive is: PR Outsourced -> NO Audit.
The phrase 'only if' introduces a necessary condition. Finding the contrapositive allows this premise to connect to the existing chain.
3
Chain all the conditional statements together.
Merger Approved -> Marketing Downsized -> PR Outsourced -> NO Audit.
The transitive property of conditional logic allows us to link shared necessary and sufficient conditions.
4
Evaluate the deduced chain to find a valid inference that matches an answer choice.
The chain directly supports the statement: If Merger Approved, then NO Audit.
In Must Be True questions, the correct answer is a direct, logically valid deduction from the provided premises.

Anahtar Kavram

Chaining Conditional Statements and Contrapositives
Tahmini Süre:1m 30s
Soru 27Soru →

Under recently enacted financial statutory guidelines, every institution acting as a primary market maker is subject to strict liquidity stress tests. Any institution subject to strict liquidity stress tests is required to hold a minimum reserve of sovereign treasury bonds. Furthermore, no institution that is required to hold a minimum reserve of sovereign treasury bonds is permitted to engage in unregulated proprietary trading. Most institutions that act as primary market makers are currently categorized by regulators as systemically important. However, it is a matter of public record that some institutions categorized as systemically important do engage in unregulated proprietary trading.

Based on the information provided above, which of the following must be true?

Cevabı ve açıklamayı göster

Cevap: At least one institution categorized as systemically important is not permitted to engage in unregulated proprietary trading.

Cevap

At least one institution categorized as systemically important is not permitted to engage in unregulated proprietary trading.
The correct answer synthesizes the conditional chain with a quantifier statement to yield a mathematically certain deduction. The stimulus establishes that all primary market makers must take stress tests, which requires holding treasury bonds, which in turn forbids unregulated proprietary trading. Therefore, all primary market makers are forbidden from unregulated proprietary trading. The text also states that most primary market makers are systemically important. Since more than half of primary market makers are systemically important, and absolutely every primary market maker is forbidden from proprietary trading, those specific overlapping institutions must be systemically important and forbidden from proprietary trading. This guarantees that at least one systemically important institution is not permitted to engage in unregulated proprietary trading.

Adım Adım Çözüm

1
Map the conditional rules applying to primary market makers (PMM).
PMM → Strict Liquidity Stress Tests (SLST) → Minimum Reserve of Treasury Bonds (MRTB) → NOT permitted to engage in Unregulated Proprietary Trading (~UPT).
To establish the full chain of mandatory constraints acting upon any institution acting as a primary market maker.
2
Simplify the conditional chain to connect the first sufficient condition to the final necessary condition.
If an institution is a primary market maker, it is categorically forbidden from engaging in unregulated proprietary trading (PMM → ~UPT).
To make the overarching relationship between the start and end variables explicit for easier synthesis with the quantifier statements.
3
Integrate the 'most' quantifier statement with the deduced conditional rule.
We are given that 'Most PMMs are Systemically Important (SI)'. Since we know absolutely all PMMs are forbidden from UPT (PMM → ~UPT), it follows that those specific PMMs that are systemically important are also forbidden from UPT.
Because the 'most' quantifier guarantees an overlap of at least one institution (more than half), and the universal rule applies to all PMMs within that overlap.
4
Formulate the final necessary deduction based on the intersection established.
There must be at least one institution that is both systemically important and forbidden from engaging in unregulated proprietary trading.
This exactly matches the logical deduction that 'some systemically important institutions are not permitted to engage in unregulated proprietary trading', which is 100% provable from the text.

Anahtar Kavram

Synthesizing conditional logic chains with existential and majoritarian quantifiers.
Soru 28Soru →

According to the city's updated zoning laws, every new building constructed in the downtown district must include a green roof. Furthermore, any building that includes a green roof is automatically eligible for a municipal tax discount. The Apex Tower is a new building that was recently constructed in the downtown district.

Based on the information above, which of the following statements must be true? Select all that apply.

Geçerli olan tümünü seçin

Cevabı ve açıklamayı göster

Cevap: The Apex Tower is eligible for a municipal tax discount.; Any new building constructed in the downtown district is eligible for a municipal tax discount.

Cevap

The statements that must be true are that the Apex Tower is eligible for a municipal tax discount, and that any new building constructed in the downtown district is eligible for a municipal tax discount.
The stimulus establishes a chain of conditional logic: New Downtown Building -> Green Roof -> Tax Discount. The correct statements perfectly track this logic by correctly linking the first condition to the last (any new downtown building gets the discount) and correctly applying it to the specific example of the Apex Tower.

Adım Adım Çözüm

1
Analyze the first conditional premise.
If a building is new and in the downtown district, then it must include a green roof.
To establish the initial condition required by the zoning laws.
2
Analyze the second conditional premise and chain it to the first.
If a building includes a green roof, then it is eligible for a tax discount. Therefore, if a building is new and in the downtown district, it is eligible for a tax discount.
To form a logical chain that connects the building's location and age directly to the tax discount.
3
Apply the factual premise to the logical chain.
The Apex Tower is a new building in the downtown district, so it must have a green roof, meaning it is eligible for a tax discount.
To determine the necessary outcomes for the specific subject mentioned in the stimulus.

Anahtar Kavram

Logical chaining of conditional statements and application to specific cases.
Soru 29Soru →

If a student takes Advanced Physics, they must have already completed Calculus. No student who has completed Calculus is eligible for the introductory math seminar. Julian is currently taking Advanced Physics.

Based on the information above, which one of the following cannot be true?

Cevabı ve açıklamayı göster

Cevap: Julian is eligible for the introductory math seminar.

Cevap

The correct answer is the statement that Julian is eligible for the introductory math seminar.
The stimulus states that taking Advanced Physics guarantees that a student has completed Calculus, and completing Calculus guarantees that a student is not eligible for the introductory math seminar. Therefore, taking Advanced Physics guarantees ineligibility for the introductory math seminar. Since Julian is taking Advanced Physics, it is a proven fact that he is not eligible for the seminar. The correct option contradicts this proven fact, making it impossible to be true.

Adım Adım Çözüm

1
Identify the first conditional relationship.
Taking Advanced Physics requires completing Calculus (Advanced Physics -> Calculus).
This establishes the necessary condition for taking Advanced Physics.
2
Identify the second conditional relationship.
Completing Calculus means a student is not eligible for the introductory math seminar (Calculus -> Not Eligible for Intro Math).
This establishes what happens once Calculus is completed.
3
Chain the conditionals together.
Advanced Physics -> Calculus -> Not Eligible for Intro Math.
Transitive logic allows us to connect the first and second statements.
4
Apply the rule to the specific case of Julian.
Because Julian is taking Advanced Physics, he must not be eligible for the introductory math seminar.
Julian triggers the sufficient condition of the chained conditionals.
5
Evaluate the options to find the one that contradicts the deduction.
The statement claiming Julian is eligible for the introductory math seminar is logically impossible.
A 'cannot be true' question asks for the option that directly violates the facts established in the stimulus.

Anahtar Kavram

Cannot Be True Questions
Soru 30Soru →

Under the new municipal zoning code, any neighborhood designated as a Transit-Oriented District must allow for high-density residential development. Furthermore, a neighborhood cannot be designated as a Transit-Oriented District unless it currently contains at least one active light-rail station. The code also specifies that any neighborhood allowing for high-density residential development is required to allocate at least 15 percent of its new housing units for low-income residents. The Historic Southside neighborhood recently had its only active light-rail station permanently decommissioned.

Which of the following must be true on the basis of the statements above?

Cevabı ve açıklamayı göster

Cevap: Any neighborhood that is not required to allocate at least 15 percent of its new housing units for low-income residents cannot be designated as a Transit-Oriented District.

Cevap

Any neighborhood that is not required to allocate at least 15 percent of its new housing units for low-income residents cannot be designated as a Transit-Oriented District.
The stimulus provides two key conditional statements that can be connected: 'Transit-Oriented District -> high-density residential' and 'high-density residential -> 15 percent low-income'. Linking these statements creates the deduction: 'Transit-Oriented District -> 15 percent low-income'. The correct answer is the logical contrapositive of this deduction: if a neighborhood is not required to allocate 15 percent for low-income housing, it cannot be a Transit-Oriented District.

Adım Adım Çözüm

1
Diagram the conditional statements provided in the stimulus.
1) TOD -> HDR
2) TOD -> ALRS (Contrapositive: ~ALRS -> ~TOD)
3) HDR -> 15% LI
4) Historic Southside -> ~ALRS
Translating text into formal logic makes it easier to track relationships and avoid traps.
2
Chain the conditional statements together where possible.
Combining premises 1 and 3 (TOD -> HDR and HDR -> 15% LI) yields the logical chain: TOD -> HDR -> 15% LI. Therefore, TOD -> 15% LI.
Connecting premises reveals hidden logical inferences that must be true.
3
Form the contrapositive of the new logical chain.
The contrapositive of (TOD -> 15% LI) is (~15% LI -> ~TOD).
Correct answers in 'Must Be True' questions are frequently the contrapositive of a chained inference.
4
Evaluate the information about the Historic Southside neighborhood.
Since Historic Southside has no active light-rail station (~ALRS), it cannot be a TOD (~TOD). No further valid deductions can be made about its high-density development or low-income housing.
This confirms that options making claims about Historic Southside's high-density development are likely testing illegal negation.

Anahtar Kavram

Conditional Logic Chaining and Contrapositives
Soru 31Soru →

The charter of the Global Botanical Society mandates that every researcher deployed to the Alpine Tundra biome complete advanced avalanche survival training. Furthermore, no individual who has completed advanced avalanche survival training is eligible to lead the primary basecamp logistics team. All scientists employed by the Northern Institute are currently deployed as researchers to the Alpine Tundra biome. Therefore, it is logically possible for a scientist employed by the Northern Institute to lead the primary basecamp logistics team.

Cevabı ve açıklamayı göster

Cevap: False

Cevap

False
The statement is false because it directly contradicts the logical deductions derived from the premises. Since all Northern Institute scientists are in the Alpine Tundra, they all must take avalanche training. Because anyone with avalanche training is barred from leading the logistics team, it is impossible for a Northern Institute scientist to hold that position.

Adım Adım Çözüm

1
Translate the rules into formal conditional logic.
Northern Institute (NI) → Alpine Tundra (AT); AT → Avalanche Training (AST); AST → NOT Logistics Team (LT).
Formalizing the rules ensures no logical connections are missed and allows for accurate chaining.
2
Chain the conditional statements together.
NI → AT → AST → NOT LT.
Chaining reveals the necessary deductions that must logically follow from the premises.
3
Evaluate the target statement against the deductions.
The target statement claims it is logically possible for an NI scientist to be on the LT. However, the chain definitively proves that all NI scientists are NOT on the LT.
A statement claiming something is possible when the premises dictate it is strictly impossible cannot be true.

Anahtar Kavram

Cannot Be True Questions
Soru 32Soru →

The mind-body problem has plagued philosophers for centuries, generating countless theoretical models. According to Dr. Varma’s latest treatise on cognitive ontology, however, most of these models are fundamentally flawed. Varma argues that a philosophical framework can successfully resolve the mind-body problem only if it discards strict substance dualism and either embraces panpsychism or redefines consciousness as an emergent physical property.

Which one of the following statements logically follows from Dr. Varma’s claims?

Cevabı ve açıklamayı göster

Cevap: If a philosophical framework fails to discard strict substance dualism, or if it neither embraces panpsychism nor redefines consciousness as an emergent physical property, then it cannot successfully resolve the mind-body problem.

Cevap

The statement indicating that a framework cannot successfully resolve the problem if it fails to discard strict substance dualism, or if it neither embraces panpsychism nor redefines consciousness.
This statement correctly applies the contrapositive and De Morgan's laws to the stimulus. The original rule is: Resolve -> Discard Dualism AND (Panpsychism OR Emergent). The contrapositive negates and reverses this: If NOT Discard Dualism, OR if NOT (Panpsychism OR Emergent), then NOT Resolve. Negating 'Panpsychism OR Emergent' results in 'NOT Panpsychism AND NOT Emergent' (which is written as 'neither... nor...').

Adım Adım Çözüm

1
Translate the rule from the stimulus into formal logic.
Resolve Mind-Body (R) -> Discard Dualism (D) AND [Embrace Panpsychism (P) OR Redefine Consciousness (E)]
The phrase 'only if' introduces a necessary condition, which in this case is a compound condition requiring 'D' alongside at least one of 'P' or 'E'.
2
Determine the logical task required by the question stem.
Formulate the contrapositive of the identified conditional rule.
The question asks what logically follows from a definitive conditional statement. In this format, the correct answer almost always represents the valid contrapositive.
3
Apply De Morgan's laws to the primary conjunction (AND).
NOT D OR NOT [P OR E] -> NOT R
To negate an 'AND' statement when forming a contrapositive, you must negate both individual components and change the 'AND' to an 'OR'.
4
Apply De Morgan's laws a second time to the nested disjunction (OR).
NOT D OR [NOT P AND NOT E] -> NOT R (meaning: If NOT D, or if neither P nor E, then NOT R).
To negate the inner 'OR' statement, you must negate both 'P' and 'E' and change the 'OR' to an 'AND'. 'NOT P AND NOT E' translates smoothly to 'neither P nor E'.

Anahtar Kavram

Forming valid contrapositives using De Morgan's laws on complex, compound conditional statements.
Tahmini Süre:2m 0s
Soru 33Soru →

Match each conditional statement regarding corporate merger regulations on the left with its logically valid contrapositive on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

The merger will be approved only if the antitrust division grants clearance and the competing firm withdraws its lawsuit.
Approval of the merger requires that the antitrust division grants clearance or the competing firm withdraws its lawsuit.
The merger is guaranteed to be approved if the antitrust division grants clearance and the competing firm withdraws its lawsuit.
The antitrust division granting clearance or the competing firm withdrawing its lawsuit is sufficient for the merger to be approved.

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

The logically valid pairs map combinations of necessary/sufficient conditions with their De Morgan's Law contrapositives. 'Approved only if... and...' pairs with 'If not... or not..., not approved'. 'Requires that... or...' pairs with 'If fails to... and refuses to..., not approved'. 'Guaranteed to be approved if... and...' pairs with 'If not approved, then either not... or not...'. '...or... is sufficient for' pairs with 'If not approved, it must be that failed to... and failed to...'.
Each correct pair properly executes the two mandatory steps of forming a complex contrapositive: (1) flipping the necessary and sufficient conditions while negating both sides, and (2) applying De Morgan's Laws to switch any negated 'AND' operator to an 'OR' operator, and vice versa.

Adım Adım Çözüm

1
Define variables for the three propositions to systematically evaluate the logical structures without relying on English syntax.
Let M = 'Merger is approved'. Let C = 'Antitrust division grants clearance'. Let W = 'Competing firm withdraws lawsuit'.
Using symbolic logic prevents confusion caused by complex phrasing and varied conditional trigger words (only if, requires, sufficient, guaranteed).
2
Translate the four left-side conditional statements into formal logic.
Statement 1: M → (C and W). Statement 2: M → (C or W). Statement 3: (C and W) → M. Statement 4: (C or W) → M.
Accurately identifying necessary versus sufficient conditions is mandatory before forming a contrapositive.
3
Form the logically valid contrapositive for each translated statement by reversing the arrow and negating both sides.
Contrapositive 1: ~(C and W) → ~M. Contrapositive 2: ~(C or W) → ~M. Contrapositive 3: ~M → ~(C and W). Contrapositive 4: ~M → ~(C or W).
The contrapositive is the only inherently valid inference that can be drawn from a standard conditional statement.
4
Apply De Morgan's Laws to distribute the negations across the compound 'AND' / 'OR' conditions.
Contrapositive 1: (~C or ~W) → ~M. Contrapositive 2: (~C and ~W) → ~M. Contrapositive 3: ~M → (~C or ~W). Contrapositive 4: ~M → (~C and ~W). These directly map to the four statements on the right.
Negating a conjunction (AND) always yields a disjunction (OR), and negating a disjunction yields a conjunction.

Anahtar Kavram

Forming Complex Contrapositives using De Morgan's Laws
Soru 34Soru →

According to the newly established directives by the Orbital Commerce Commission, a commercial satellite launch qualifies for priority scheduling only if the launching corporation submits a comprehensive debris mitigation plan and at least one of the following occurs: the payload serves a registered humanitarian function, or an independent aerospace auditor certifies the vehicle's structural integrity.

Which of the following statements must be logically true based on the directives above?

Cevabı ve açıklamayı göster

Cevap: If a launching corporation fails to submit a comprehensive debris mitigation plan, or if both the payload does not serve a registered humanitarian function and an independent aerospace auditor fails to certify the vehicle's structural integrity, then the commercial satellite launch will not qualify for priority scheduling.

Cevap

If a launching corporation fails to submit a comprehensive debris mitigation plan, or if both the payload does not serve a registered humanitarian function and an independent aerospace auditor fails to certify the vehicle's structural integrity, then the commercial satellite launch will not qualify for priority scheduling.
The correct answer accurately applies the contrapositive to a compound conditional. The stimulus dictates that for a launch to qualify, it needs a mitigation plan AND at least one of two other factors (humanitarian function OR auditor certification). If it lacks the mitigation plan, it fails. Alternatively, if it lacks BOTH of the other factors, it fails. The correct choice perfectly captures this dynamic.

Adım Adım Çözüm

1
Deconstruct the stimulus into formal conditional logic.
Priority Scheduling (P) → Mitigation Plan (M) AND [Humanitarian Function (H) OR Auditor Certification (A)].
Accurately representing the necessary conditions is required before attempting to find the contrapositive.
2
Apply the process of forming a contrapositive by reversing and negating the terms.
NOT {Mitigation Plan (M) AND [Humanitarian Function (H) OR Auditor Certification (A)]} → NOT Priority Scheduling (P).
The logically valid inference from any conditional statement is its contrapositive.
3
Apply De Morgan's laws to properly negate the compound necessary condition.
The negation of 'M AND (H OR A)' becomes 'NOT M OR NOT (H OR A)'. Applying De Morgan's again to the inner bracket yields 'NOT M OR (NOT H AND NOT A)'.
Negating an 'AND' requires switching it to an 'OR', and negating an 'OR' requires switching it to an 'AND'.
4
Translate the final logical notation back into natural language.
If the corporation does not submit a plan (NOT M), OR if the payload lacks a humanitarian function AND the auditor fails to certify (NOT H AND NOT A), then the launch does not get priority scheduling (NOT P).
The correct answer choice will mirror this exact logical relationship without confusing necessary and sufficient conditions.

Anahtar Kavram

Forming contrapositives of compound conditional statements using De Morgan's laws.
Soru 35Soru →

All amphibians that inhabit the alpine tundra rely on the nectar of the *Saxifraga* plant during their larval stage. Most species that rely on *Saxifraga* nectar during their larval stage are capable of surviving sub-zero temperatures. Furthermore, any species capable of surviving sub-zero temperatures possesses elevated levels of blood glucose.

Based on the information provided, which of the following must be true? (Select all that apply)

Geçerli olan tümünü seçin

Cevabı ve açıklamayı göster

Cevap: Most species that rely on the nectar of the *Saxifraga* plant during their larval stage possess elevated levels of blood glucose.; Any species that does not possess elevated levels of blood glucose is not capable of surviving sub-zero temperatures.

Cevap

The correct statements are that most species relying on the nectar possess elevated blood glucose, and that any species lacking elevated blood glucose cannot survive sub-zero temperatures.
The correct statements are derived from valid logical rules. One is proven by chaining a 'most' premise directly into an 'all' premise, establishing that most nectar-reliant species possess elevated blood glucose. The other is proven by correctly forming the contrapositive of the final premise, establishing that lacking elevated blood glucose guarantees a species cannot survive sub-zero temperatures.

Adım Adım Çözüm

1
Diagram the formal logic presented in the stimulus.
1. Alpine Tundra Amphibians -> Rely on Saxifraga Nectar. 2. Rely on Saxifraga Nectar -[most]-> Survive Sub-zero Temperatures. 3. Survive Sub-zero Temperatures -> Elevated Blood Glucose.
Mapping conditional and quantified statements clarifies the relationships and allowable deductions.
2
Evaluate combined logical inferences based on the diagram.
By linking statements 2 and 3, we can deduce that since most nectar-reliant species survive sub-zero temperatures, and ALL species that survive sub-zero temperatures possess elevated blood glucose, most nectar-reliant species must possess elevated blood glucose.
Chaining a 'most' statement with an 'all' statement yields a valid 'most' deduction.
3
Form contrapositives of the absolute conditional statements.
Contrapositive of 1: Do not rely on Saxifraga nectar -> Not an Alpine Tundra Amphibian. Contrapositive of 3: No elevated blood glucose -> Cannot survive sub-zero temperatures.
Contrapositives are always logically equivalent to the original conditional statements and are frequently tested in Must Be True questions.
4
Evaluate the given choices against the deduced truths to identify the correct answers.
The choice stating most nectar-reliant species possess elevated blood glucose matches our chained inference. The choice stating species without elevated blood glucose cannot survive sub-zero temperatures matches the contrapositive of statement 3.
Comparing choices to mathematically proven deductions ensures we select only the statements that are 100% supported by the text.

Anahtar Kavram

Chaining formal logic and forming contrapositives to determine valid deductions.
Soru 36Soru →

According to the planetary geology archives:

- Every meteorite classified as a Type C chondrite contains elevated levels of iridium.
- Any meteorite containing elevated levels of iridium is designated as a Class-1 research priority by the Astromaterials Board.
- No meteorite recovered from the Antarctic glacial surface is designated as a Class-1 research priority unless it is a newly discovered fragment that has not been previously cataloged.
- Some meteorites classified as Type C chondrites were recovered from the Antarctic glacial surface, and most of these are exceptionally well-preserved.

Based on the information provided, which of the following statements must be true? (Select all that apply.)

Geçerli olan tümünü seçin

Cevabı ve açıklamayı göster

Cevap: Every Type C chondrite recovered from the Antarctic glacial surface is a newly discovered fragment.; At least some exceptionally well-preserved meteorites contain elevated levels of iridium.

Cevap

The correct statements are that every Type C chondrite recovered from the Antarctic glacial surface is a newly discovered fragment, and that at least some exceptionally well-preserved meteorites contain elevated levels of iridium.
Based on the rules, any Type C chondrite is a Class-1 priority. Therefore, any Type C chondrite from Antarctica is both from Antarctica and a Class-1 priority, which means it must be a newly discovered fragment. Additionally, because some Type C chondrites from Antarctica exist, and most of them are well-preserved, there is at least one well-preserved Type C chondrite. Since it is Type C, it must contain elevated levels of iridium.

Adım Adım Çözüm

1
Map the universal conditional statements.
Type C -> Iridium -> Class-1 Priority. (Antarctic AND Class-1) -> Newly Discovered.
This establishes the strict logical chains required to evaluate absolute statements.
2
Analyze the quantified statements ('some' and 'most').
There exists a group of meteorites that are Type C and Antarctic. Most of this specific group are well-preserved.
Quantifiers provide the existential facts that interact with the universal rules.
3
Combine the existential facts with the universal chains.
The (Type C + Antarctic) meteorites must be Class-1 (since they are Type C). Because they are (Class-1 + Antarctic), they must be newly discovered. Thus, all (Type C + Antarctic) are newly discovered.
This guarantees the truth of the statement regarding Type C chondrites from Antarctica.
4
Derive inferences from the 'most' statement.
Since 'most' of the (Type C + Antarctic) group are well-preserved, at least one such meteorite exists. Because this meteorite is Type C, it contains iridium. Therefore, some well-preserved meteorites contain iridium.
This guarantees the truth of the statement regarding well-preserved meteorites.

Anahtar Kavram

Chaining Conditional Statements and Quantifier Logic
Soru 37Soru →

According to the recent interstellar survey of the Orionis sector:

All rogue planets detected in the outer ring possess a dense atmosphere of frozen methane. Any planet with a dense atmosphere of frozen methane reflects less than 5% of incident starlight. Most rogue planets detected in the outer ring have a core temperature below -200 degrees Celsius. Furthermore, no planet that reflects less than 5% of incident starlight is currently targeted for deep-core drilling.

Based strictly on the information above, which of the following statements must be true? (Select all valid deductions)

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Cevabı ve açıklamayı göster

Cevap: Every rogue planet detected in the outer ring is currently excluded from deep-core drilling targets.; Any planet that is targeted for deep-core drilling reflects 5% or more of incident starlight.

Cevap

The correct deductions are that every rogue planet in the outer ring is excluded from deep-core drilling targets, and that any planet targeted for deep-core drilling reflects 5% or more of incident starlight.
The valid deductions rely on strict conditional logic. By chaining the premises, we find that being an outer ring rogue planet guarantees a frozen methane atmosphere, which guarantees reflecting less than 5% of starlight, which in turn guarantees exclusion from drilling targets. Therefore, any such rogue planet is excluded. Additionally, the contrapositive of the final premise dictates that if a planet IS targeted for drilling, it cannot reflect less than 5% of starlight (meaning it must reflect 5% or more).

Adım Adım Çözüm

1
Map the primary conditional chain provided in the stimulus.
Rogue Planet in Outer Ring (RP) -> Frozen Methane (FM) -> Reflects < 5% Light (<5L).
This establishes the basic transitive relationship from the first two premises.
2
Add the final conditional premise to the chain.
RP -> FM -> <5L -> NOT Targeted for Drilling (~TD).
The final premise introduces a negative necessary condition based on the light reflection.
3
Evaluate the beginning and end of the completed conditional chain.
RP -> ~TD (If a planet is a rogue planet in the outer ring, it is not targeted for drilling).
This proves that the statement claiming every rogue planet in the outer ring is excluded from drilling targets must be logically true.
4
Form the contrapositive of the relationship between light reflection and drilling targets.
The original rule is: If <5L -> ~TD. The contrapositive is: If TD -> NOT <5L (meaning 5% or more).
This proves that the statement claiming any targeted planet reflects 5% or more starlight must be logically true.

Anahtar Kavram

Transitive conditional reasoning and contrapositive formation
Soru 38Soru →

Consider the following premises about an ecosystem:

All migratory birds that nest in the Alpine Ridge rely on the silver-leaf willow for food.
Most migratory birds that nest in the Alpine Ridge have developed a resistance to the local avian flu.
Every bird that relies on the silver-leaf willow for food has a beak specially adapted for cracking seeds.
Most birds that have developed a resistance to the local avian flu carry a specific genetic marker, GM-42.

Based on these premises, it must be true that at least some birds with beaks specially adapted for cracking seeds carry the GM-42 genetic marker.

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement could be false, so it does not logically have to be true.
The statement is logically invalid because mathematically connecting 'Some A are B' with 'Most B are C' does not guarantee 'Some A are C'. The 'most' group of B could entirely exclude the 'some' group of A.

Adım Adım Çözüm

1
Deduce the relationship between Alpine Ridge birds and beak adaptations.
All Alpine Ridge birds rely on the willow, and all willow-reliant birds have adapted beaks. Therefore, all Alpine Ridge birds have adapted beaks (A → B, B → C means A → C).
This establishes a formal necessary condition for all Alpine Ridge birds, simplifying the first part of the stimulus.
2
Integrate the 'most' premise regarding Alpine Ridge birds and resistance.
Since most Alpine Ridge birds are resistant, and all Alpine Ridge birds have adapted beaks, we can validly deduce that at least some birds with adapted beaks are resistant (Some A are B).
This maps the relationship between the two target variables presented in the conclusion: adapted beaks and flu resistance.
3
Evaluate the final link between flu resistance and the GM-42 genetic marker.
The premises yield two isolated relationships: 'Some birds with adapted beaks are resistant' and 'Most resistant birds have the GM-42 marker'.
Isolating these formal relationships allows us to test the proposed conclusion strictly against the rules of quantifier logic.
4
Test the validity of combining 'Some X are Y' with 'Most Y are Z'.
There is no mathematical guarantee that the 'some' overlap in the first relationship shares any members with the 'most' majority in the second relationship. Thus, no valid deduction can be made.
This proves the target statement is a logical overreach and could be false.

Anahtar Kavram

Formal deductions from overlapping quantifiers (the invalidity of chaining 'Some' and 'Most')
Soru 39Soru →

Evaluate the logical validity of the deduction based on the provided premises.

Premises:
1. All ceremonial masks recovered from the archaeological site are adorned with lapis lazuli.
2. Most of the artifacts adorned with lapis lazuli recovered from the site date back to the Second Dynasty.

Deduction: Based on these premises, it must be true that most of the ceremonial masks recovered from the site date back to the Second Dynasty.

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The deduction is false. We cannot logically conclude that most of the ceremonial masks date back to the Second Dynasty.
The correct evaluation is false because the logical chain 'All A are B' and 'Most B are C' does not yield 'Most A are C'. The ceremonial masks are a subset of the lapis lazuli artifacts. It is possible that the lapis lazuli artifacts from the Second Dynasty consist entirely of non-mask objects, meaning zero masks might actually date to the Second Dynasty.

Adım Adım Çözüm

1
Translate the premises into formal quantifier logic structures.
Let A = ceremonial masks, B = artifacts adorned with lapis lazuli, and C = artifacts dating to the Second Dynasty. Premise 1: All A are B. Premise 2: Most B are C.
Translating natural language into categorical relationships prevents relying on outside assumptions.
2
Determine the necessary overlap between the specific subgroup (A) and the characteristic (C).
Because 'A' is merely a subset of 'B', and only 'most' (not all) of 'B' are 'C', the subset 'A' could entirely exist within the minority portion of 'B' that is NOT 'C'.
To test if the 'most' quantifier logically guarantees a trait for every contained subset.
3
Evaluate the proposed deduction.
The deduction claims that 'Most A are C' must be true. Since it is entirely possible that zero 'A' are 'C', the deduction is invalid.
In formal logic, a deduction is only valid if it MUST be true under all scenarios permitted by the premises.

Anahtar Kavram

Quantifier Logic: Invalid chaining of 'All' and 'Most'
Soru 40Soru →

In the early eighteenth century, national copyright statutes were enacted primarily to regulate the book trade, protecting publishers from the financial ruin of unauthorized mass printing. By the late nineteenth century, however, a cultural shift had elevated the concept of the "original genius" of the creator. Consequently, courts began interpreting these same copyright statutes as mechanisms for protecting an author's unique creative expression. Interestingly, throughout this period of evolving judicial interpretation, the actual legislative text of the copyright statutes remained almost entirely unchanged, retaining its original focus on the mechanics of printing and distribution.

Which one of the following is most strongly supported by the information above?

Cevabı ve açıklamayı göster

Cevap: Late nineteenth-century courts granted protections under the copyright statutes that differed from the primary protections those statutes were enacted to provide.

Cevap

Late nineteenth-century courts granted protections under the copyright statutes that differed from the primary protections those statutes were enacted to provide.
The stimulus explicitly states that the copyright statutes were originally enacted primarily to protect publishers. It then notes that by the late nineteenth century, courts used these same, unchanged statutes to protect authors' creative expression. This directly supports the inference that the courts were using the statutes to provide protections that differed from their original, primary legislative purpose.

Adım Adım Çözüm

1
Identify the original primary purpose of the statutes.
The statutes were enacted in the early eighteenth century primarily to regulate the book trade and protect publishers.
This establishes the baseline intent and function of the legislation.
2
Identify how the interpretation of these statutes changed by the late nineteenth century.
Courts began interpreting them as mechanisms for protecting an author's unique creative expression.
This shows a shift in the application of the law driven by cultural changes.
3
Determine if the actual laws changed to reflect this new interpretation.
The legislative text remained almost entirely unchanged.
This confirms that the new protections were extracted from the old text.
4
Synthesize the findings to formulate a supported inference.
It can be safely concluded that courts were granting protections (for authors) that were fundamentally different from the protections the statutes were originally enacted to provide (for publishers).
This directly connects the premises without assuming outside information.

Anahtar Kavram

Synthesizing factual premises to derive a safe, provable inference without relying on outside assumptions.
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