Question 1

Read the following syllogistic statements and conclusions, and answer the question: Statements: I. All cups are plates. II. No plate is a shop. Conclusions: I. No cup is a shop. II. No shop is a plate. Which of the given conclusions logically follow…

Accepted Answer

Introduction / Context: This question is a standard syllogism problem. You are given two categorical statements about sets (cups, plates and shops) and two candidate conclusions. You must decide which conclusions follow logically, assuming the statements are absolutely true, even if they conflict with common sense or real life. Syllogism questions test your ability to reason about subset and disjoint-set relationships. Given Data / Assumptions: • Statement I: All cups are plates. (Every cup belongs to the set of plates.) • Statement II: No plate is a shop. (The sets plates and shops are completely disjoint.) • Conclusion I: No cup is a shop. • Conclusion II: No shop is a plate. • We must accept the statements as true and check which conclusions must also be true in all possible diagrams that satisfy the statements. Concept / Approach: Syllogisms can be visualised using Venn diagrams. "All cups are plates" means the cup circle lies completely inside the plate circle. "No plate is a shop" means the plate circle and the shop circle do not intersect at all. From these, we check whether the proposed conclusions necessarily hold in every such configuration. Step-by-Step Solution: Step 1: Represent Statement I: All cups are plates. So the set Cup is a subset of Plate. Step 2: Represent Statement II: No plate is a shop. So Plate and Shop are disjoint sets; they have no overlap. Step 3: Test Conclusion I: No cup is a shop. Since all cups are inside the Plate set and Plate does not overlap with Shop, it is impossible for any cup to be in Shop. Therefore, Conclusion I must be true. Step 4: Test Conclusion II: No shop is a plate. Statement II "No plate is a shop" is logically equivalent to "No shop is a plate", because "no A is B" is symmetrical: if there is no object that is both A and B, then equally there is no object that is both B and A. Hence Conclusion II also must be true. Verification / Alternative check: Draw three sets: Cups inside Plates, and Shops completely separate from Plates. You will see instantly that the Cup set is also separate from the Shop set, confirming Conclusion I. Also, no element of Shops can be in Plates, which restates Conclusion II. Any attempt to violate either conclusion would force a contradiction with the given statements. Why Other Options Are Wrong: Option A and Option B each keep only one of the two conclusions, but as we have seen both are valid. Option D says that neither conclusion follows, which is clearly false because both conclusions directly reflect the given set relations. Common Pitfalls: Learners sometimes forget that "No plate is a shop" automatically means "No shop is a plate" as well, and they treat them as different. Others overcomplicate the diagram or think that because cups are not mentioned in Statement II, nothing can be said about them and shops. Always track how subsets and disjoint sets interact. Final Answer: Both conclusions are logically forced by the statements, so both conclusions I and II follow.

Question 2

The district administration has issued a circular to all farmers under its jurisdiction, advising them not to use pesticides indiscriminately because such use may pollute the ground water. Which of the following can be reasonably assumed from this s…

Accepted Answer

Introduction / Context: This is a statement-and-assumption question from logical reasoning. The administration warns farmers not to use pesticides indiscriminately because it may pollute ground water. Your task is to identify which hidden belief the administration must be taking for granted when issuing this advisory. Assumptions are unstated ideas that must be true for the statement to be meaningful or purposeful. Given Data / Assumptions: • Statement: The district administration advises farmers not to use pesticides indiscriminately, as this may pollute ground water. • Option A suggests that people may stop using ground water if pollution continues. • Option B suggests that polluted ground water can cause health problems. • Option C suggests that farmers will definitely change their behaviour. • Option D suggests that most people depend on ground water for drinking. Concept / Approach: An assumption is implicit when, without it, the main statement would lose its force or purpose. Here, the administration is concerned about ground water pollution due to pesticide use. We must ask: why would the administration be concerned about ground water being polluted at all? Typically, such concern is driven by the impact on human health and safety. This gives us a strong clue toward the correct assumption. Step-by-Step Solution: Step 1: Focus on the link "pesticides indiscriminately" → "pollute the ground water". The administration warns against this because pollution of ground water is considered bad. Step 2: For ground water pollution to be a serious issue, the administration must assume that polluted ground water has negative consequences, especially for the health of people who use it. Step 3: Option B directly expresses this: polluted ground water can cause health problems. This explains why the administration is worried enough to issue a circular. Step 4: Option A talks about people stopping the use of ground water; this might or might not happen and is not required for issuing the circular. The administration wants to prevent pollution; it does not depend on people already planning to stop using ground water. Step 5: Option C assumes a definite behavioural change by farmers, but the circular is an attempt to influence behaviour, not a guarantee that they will change. It is an intention, not an assumption. Step 6: Option D suggests heavy dependence on ground water, which may be likely but is not strictly necessary. Even if dependence were moderate, polluted ground water could still be a serious health concern. Verification / Alternative check: If Option B were false, meaning polluted ground water did not create health problems, the circular would lack a strong practical reason. Why warn farmers at all if pollution did not hurt people or the environment? On the other hand, even if A, C or D are false, the administration could still sensibly warn farmers because health risks exist. This confirms that only B is essential. Why Other Options Are Wrong: Options A and D are about possible reactions or usage patterns of people; they are not necessary for the administration to issue the warning. Option C confuses the hoped-for outcome with the underlying assumption. The administration does not assume farmers will definitely obey; it only recommends that they should. Common Pitfalls: Many candidates think any statement that sounds reasonable is an assumption. Instead, ask: if this were false, would the main statement collapse logically? Only such statements are genuine assumptions. Also, avoid reading extra information into the statement, such as detailed statistics about ground water usage, which are not mentioned. Final Answer: The most reasonable and necessary assumption is that polluted ground water can cause health problems, so Option B is implicit.

Question 3

In each of the following questions, two statements (I and II) are given. They may stand in a cause-and-effect relationship, or they may be effects of the same or different causes, or even completely independent. Read the following pair and answer ac…

Accepted Answer

Introduction / Context: This problem asks you to identify the cause-and-effect relationship between two real life statements involving a university, its colleges and a mobile phone ban. Cause-and-effect questions test whether you can recognise which event logically explains the occurrence of another, rather than simply occurring near it in time. Given Data / Assumptions: • Statement I: The university instructs all affiliated colleges to ban the use of mobile phones inside the college premises. • Statement II: Most teachers signed a joint petition to the university complaining about disturbances, presumably in classrooms where teaching is affected. • We assume the university responds to issues raised by staff and that classes can be disturbed by mobile phone usage by students and possibly others. Concept / Approach: A cause is an event or situation that directly or plausibly leads to another event called the effect. Here, we must decide whether the teachers complaints led to the university ban, or whether the ban itself caused the teachers to complain. We check which direction fits more naturally with administrative decision making in colleges. Step-by-Step Solution: Step 1: Consider Statement II as the starting point. Teachers are experiencing disturbances in class, likely due to frequent or inappropriate mobile phone use by students. They collectively sign a petition to the university asking for action. Step 2: The university, after receiving this formal complaint from the majority of teachers, decides to issue a circular instructing all colleges to ban mobile phones inside the premises. Step 3: This chain of events is natural: first problem and complaint (cause), then policy decision (effect). Step 4: Reverse the direction as a test. If the university unilaterally banned mobile phones first (Statement I) and then teachers complained about disturbances (Statement II), it would not make sense, because a ban should reduce disturbances, not cause them. So Statement I is unlikely to be the cause of Statement II. Step 5: Therefore, it is most logical to treat Statement II as the cause and Statement I as the resulting effect. Verification / Alternative check: Ask whether both statements could simply be independent effects of some other cause. For example, a general rise in mobile misuse might prompt both teacher complaints and university actions independently. However, the specific wording "joint petition to the university" strongly suggests a direct link: the university is reacting to this petition. Hence, the simplest and strongest explanation is that Statement II causes Statement I. Why Other Options Are Wrong: Option A reverses the direction and implies the ban caused the petition, which conflicts with normal institutional behaviour. Option C and Option D suggest independence, which ignores the explicit connection implied by a complaint to the same authority that issued the ban. Therefore, they do not fit the situation as well as Option B does. Common Pitfalls: A typical mistake is to confuse temporal order with causal order. Just because one statement is listed first does not mean it is the cause. Always check which event naturally explains why an authority or group would take action. In governance and education systems, petitions and complaints often precede policy decisions. Final Answer: Thus, the majority teachers petition (Statement II) is the cause and the university's mobile ban (Statement I) is its effect, so Statement II is the cause and Statement I is its effect.

Question 4

The district administration has issued a circular to all farmers under its jurisdiction, advising them not to use pesticides indiscriminately because such use may pollute the ground water. Which of the following can be reasonably assumed from this s…

Accepted Answer

Introduction / Context: This question repeats the same logical pattern about pesticides and ground water. The district administration warns farmers that indiscriminate pesticide use may pollute ground water. You must identify which assumption the administration must be making about the consequences of such pollution, even though that assumption is not stated explicitly. Given Data / Assumptions: • Official circular: Farmers are advised not to use pesticides indiscriminately because this may pollute ground water. • Option A: People may stop using ground water if pollution continues. • Option B: Polluted ground water can cause health problems. • Option C: Farmers will certainly stop indiscriminate use of pesticides. • Option D: Most people in the district rely on ground water for drinking. Concept / Approach: In statement-and-assumption questions, we ask what must be true in the background for the speaker to be worried or motivated enough to make the statement. The administration is concerned about ground water pollution. Usually such concern is tied to human health and safety, rather than to any minor or cosmetic issue. Step-by-Step Solution: Step 1: The reason mentioned in the circular is that pesticides may pollute ground water. Pollution in itself is neutral; its seriousness depends on the consequences. Step 2: For this to be a strong reason to advise farmers, the administration must assume that polluted ground water is harmful, especially to human health. Step 3: Option B directly states that polluted ground water can cause health problems. This fits perfectly as the underlying concern. Step 4: Option A suggests how people might react in the future, but the administration does not need to assume that people will stop using ground water; it is enough that continuing to use polluted water would be dangerous. Step 5: Option C confuses an intended outcome with an assumption. Authorities hope farmers will change behaviour, but they cannot assume it as a fact while drafting a circular. Step 6: Option D speaks about how many people depend on ground water. Even if dependence is moderate, polluted water can still be dangerous. So heavy dependence is not strictly required as an assumption. Verification / Alternative check: If Option B were false and polluted ground water had no health impact, the circular would be pointless. Authorities would not worry about such pollution. By contrast, if A, C or D are false, the circular still makes sense as a preventive measure: any degree of harmful pollution is worth warning against. This shows that B is the only necessary assumption. Why Other Options Are Wrong: Options A and D deal with possible behaviour or usage patterns that are not required for the warning to be reasonable. Option C assumes guaranteed compliance, which is unrealistic and not necessary for issuing advice. The circular is motivated by risk, not by certainty about behaviour or dependency patterns. Common Pitfalls: Candidates often mislabel predictions and hopes as assumptions. Remember that assumptions are background facts that must hold for the reasoning to make sense, not outcomes that the speaker desires. Also, do not assume extreme dependence on a resource unless the statement itself points toward that detail. Final Answer: The only assumption that must be true for the circular to have force is that polluted ground water can cause health problems, so Option B is implicit.

Question 5

In the following question, two statements and two conclusions are given. Consider the statements to be true, even if they appear to contradict commonly known facts, and decide which conclusion(s) logically follow: Statement I: All cells are chargers…

Accepted Answer

Introduction / Context: This is a syllogism question involving three sets: cells, chargers and batteries. You are given two universal statements and asked whether each of two proposed conclusions follows logically. Syllogisms like this test your ability to track subset relationships and infer new relationships between categories. Given Data / Assumptions: • Statement I: All cells are chargers. (Cells ⊆ Chargers.) • Statement II: All batteries are cells. (Batteries ⊆ Cells.) • Conclusion I: Some chargers are batteries. • Conclusion II: All batteries are chargers. • Assume that the sets mentioned are non empty, which is standard in many exam level syllogism questions when talking about real world objects like batteries and cells. Concept / Approach: We translate each statement into set notation and form chains of inclusion. "All batteries are cells" means every battery is inside the set of cells. "All cells are chargers" means every cell is inside the set of chargers. Combining these two statements gives us a chain from batteries to chargers through cells. From that, we test each conclusion. Step-by-Step Solution: Step 1: From Statement II, Batteries ⊆ Cells. Step 2: From Statement I, Cells ⊆ Chargers. Step 3: Combine these: if Batteries ⊆ Cells and Cells ⊆ Chargers, then Batteries ⊆ Chargers. This directly yields that all batteries are chargers, which is Conclusion II. Step 4: Therefore, Conclusion II clearly follows from the given statements. Step 5: Now test Conclusion I, which says "Some chargers are batteries." If all batteries are chargers and there is at least one battery, then that battery is also a charger. Hence, there exists at least one object that is both a charger and a battery, so "Some chargers are batteries" holds. Step 6: Under the usual exam assumption that categories like batteries are not empty, Conclusion I also follows. Verification / Alternative check: Imagine a diagram: draw three nested sets. The innermost set is Batteries. Around it, a larger circle called Cells. Around that, an even larger circle called Chargers. In this picture, every battery is a cell, and every cell is a charger. Clearly, all batteries are chargers (Conclusion II). Also, since at least one battery exists, that battery lies inside the Chargers set, so some chargers are indeed batteries (Conclusion I). Why Other Options Are Wrong: Option A (only I follows) is wrong because it ignores the stronger relationship that all batteries are chargers. Option B (only II follows) is incomplete in the usual exam context where categories are assumed non empty. Option D (neither follows) contradicts the obvious subset chain Batteries ⊆ Cells ⊆ Chargers. Common Pitfalls: Some students strictly adopt a no existence rule for universal statements and argue that "All X are Y" does not guarantee that any X actually exists, which would block conclusions starting with "Some". However, in most competitive exams about real objects, the existence of at least one member is implicitly assumed. Others make the mistake of reversing the subset and think that some chargers must be cells or that all chargers are batteries, which is not given. Final Answer: On the usual exam interpretation, both conclusions I and II follow from the given statements.

Question 6

In the following question, two symbolic statements are given: Statements: (I) a R b (II) b R a Conclusions: I. The relation R is symmetric. II. R is a relation (on some set). Assuming the two statements about a and b are true, which of the given con…

Accepted Answer

Introduction / Context: This is a reasoning question about the properties of a relation R. You are told that a R b and b R a are both true and asked whether you can conclude that R is symmetric and that it is a relation. The question tests your understanding of definitions in discrete mathematics and how much can be inferred from a small amount of information. Given Data / Assumptions: • Statement (I): a R b is true for some elements a and b. • Statement (II): b R a is true for the same pair of elements. • Conclusion I: The relation R is symmetric. • Conclusion II: R is a relation (on some underlying set). • A symmetric relation is usually defined on a set S such that for all x and y in S, if x R y then y R x. Concept / Approach: A relation R on a set is simply a set of ordered pairs (x, y) where x and y belong to some underlying set. From the given statements, we know that the ordered pairs (a, b) and (b, a) are in R. We ask: does this limited information prove that R is symmetric for all possible pairs, or only shows that R is at least some relation that happens to contain these two pairs? Step-by-Step Solution: Step 1: From a R b and b R a, we can confidently assert that R is a relation that contains at least two ordered pairs: (a, b) and (b, a). So it is meaningful to speak of R as a relation. Step 2: Check Conclusion II: "This is a relation." Since R is already used with ordered pairs, it meets the basic definition of a relation, so Conclusion II follows. Step 3: Now examine Conclusion I: "This relation is symmetric." Symmetry requires that for every x and y in the underlying set, if x R y holds then y R x must also hold. However, we only know about one specific pair (a, b). We do not know what happens for any other pair (x, y) different from (a, b). Step 4: It is possible that R includes (a, b) and (b, a) but fails symmetry for some other elements, for example (c, d) is in R but (d, c) is not. The given information does not exclude such possibilities. Step 5: Therefore, the data is insufficient to conclude that R is symmetric overall. We only know that symmetry holds for this one pair a and b. Verification / Alternative check: Construct a counterexample. Let the underlying set be {a, b, c}. Define R = {(a, b), (b, a), (c, a)}. Here a R b and b R a are both true, but c R a is true while a R c is false. So R is not symmetric as a whole. Since this example satisfies the given statements yet violates symmetry, Conclusion I cannot be guaranteed by the given data. Why Other Options Are Wrong: Option A claims only symmetry follows, which we have shown is not justified. Option C claims that no conclusion follows, but we clearly know R is a relation because we are given two valid instances of it. Option D claims that both conclusions follow, which again wrongly asserts symmetry from insufficient information. Common Pitfalls: A common error is to confuse "R is symmetric for a specific pair a and b" with "R is symmetric as a property for all pairs". One example does not prove a universal property. Always check whether you are being asked to prove a general rule or just to check a specific instance. Final Answer: Only the fact that R is a relation follows logically, so only Conclusion II is valid.

Statement and Assumption Questions