Daily Olympiad: Logical Reasoning - Syllogisms [20260716]

Solution

Correct: C

Let's represent the statements using Venn diagrams: Statement 1: All pens are pencils. (Pens circle is entirely inside Pencils circle) Statement 2: Some pencils are erasers. (Erasers circle overlaps with a portion of the Pencils circle). Now, let's analyze the conclusions: I. Some pens are erasers. From the diagram, it's possible that the Erasers overlap with the Pens, but it's not certain. The overlap could be only with the Pencils that are not Pens. So, Conclusion I is not necessarily true (uncertain). II. No pen is an eraser. Similarly, it's possible that Erasers do not overlap with Pens at all, but it's not certain. So, Conclusion II is also not necessarily true (uncertain). However, conclusions I (Some pens are erasers) and II (No pen is an eraser) form a complementary pair (specifically, a contradictory pair 'I' type and 'E' type propositions). If 'Some A are B' is false, then 'No A are B' must be true, and vice-versa. Since both are individually uncertain, one of them must be true. Therefore, 'Either conclusion I or II follows'.

Solution

Correct: A

Let's represent the statements: Statement 1: Some tables are chairs. (Overlap between Tables and Chairs circles). Statement 2: All chairs are furniture. (Chairs circle is entirely inside Furniture circle). Combining the statements: Since 'Some tables are chairs' and 'All chairs are furniture', it logically follows that 'Some tables are furniture'. Now, let's analyze the conclusions: I. Some furniture are tables. This is a conversion of 'Some tables are furniture', which we derived from the premises. If 'Some A are B', then 'Some B are A' is a valid conversion. Thus, Conclusion I follows. II. All tables are furniture. From the statements, we only know that 'Some tables are furniture'. It is possible that some tables are not furniture (e.g., if there are tables that are not chairs, and those are not furniture). So, Conclusion II does not necessarily follow. Therefore, only conclusion I follows.

Solution

Correct: B

Let's represent the statements: Statement 1: No student is a teacher. (Students circle and Teachers circle are separate). Statement 2: All teachers are intelligent. (Teachers circle is entirely inside Intelligent circle). Now, let's analyze the conclusions: I. No student is intelligent. From the statements, we know that students are not teachers, and teachers are intelligent. However, this does not mean that students cannot be intelligent. There could be intelligent people who are not teachers but are students. So, Conclusion I does not necessarily follow. II. Some intelligent are not students. Since 'All teachers are intelligent' (Statement 2) and 'No student is a teacher' (Statement 1), it implies that the group of 'teachers' (who are also 'intelligent') cannot be 'students'. Therefore, 'Some intelligent (specifically, the teachers who are intelligent) are not students' is a valid conclusion. Thus, Conclusion II follows. Therefore, only conclusion II follows.

Solution

Correct: E

Let's represent the statements: Statement 1: All birds are animals. (Birds circle is entirely inside Animals circle). Statement 2: All animals are mammals. (Animals circle is entirely inside Mammals circle). Combining the statements: Since 'All birds are animals' and 'All animals are mammals', it logically follows that 'All birds are mammals'. Now, let's analyze the conclusions: I. Some birds are mammals. If 'All A are B' is true, then 'Some A are B' must also be true. Since 'All birds are mammals' is true, 'Some birds are mammals' is also true. Thus, Conclusion I follows. II. All birds are mammals. This conclusion directly follows from the combination of the two statements. Thus, Conclusion II follows. Therefore, both conclusions I and II follow.

Solution

Correct: A

Let's represent the statements: Statement 1: Some fruits are sweet. (Overlap between Fruits and Sweet circles). Statement 2: All sweet things are healthy. (Sweet circle is entirely inside Healthy circle). Combining the statements: Since 'Some fruits are sweet' and 'All sweet things are healthy', it logically follows that the portion of fruits that are sweet must also be healthy. Therefore, 'Some fruits are healthy'. Now, let's analyze the conclusions: I. Some fruits are healthy. This conclusion directly follows from the combination of the two statements. Thus, Conclusion I follows. II. All healthy things are fruits. We only know that 'Some fruits are healthy'. It is possible for healthy things to exist that are not fruits (e.g., vegetables, exercise). So, Conclusion II does not necessarily follow. Therefore, only conclusion I follows.

Solution

Correct: A

Let's represent the statements: Statement 1: Some shirts are pants. (Overlap between Shirts and Pants circles). Statement 2: No pants are socks. (Pants circle and Socks circle are separate). Now, let's analyze the conclusions: I. Some shirts are not socks. From Statement 1, there is a portion of shirts that are pants. From Statement 2, no pants are socks. Therefore, that specific portion of shirts (which are pants) cannot be socks. This implies 'Some shirts are not socks'. Thus, Conclusion I follows. II. No shirts are socks. While the part of shirts that are pants are not socks, there might be other shirts (not pants) that could potentially be socks. The statements do not provide enough information to conclude that absolutely no shirts are socks. So, Conclusion II does not necessarily follow. Therefore, only conclusion I follows.

Solution

Correct: D

Let's represent the statements: Statement 1: All cars are vehicles. (Cars circle is entirely inside Vehicles circle). Statement 2: Some vehicles are trucks. (Trucks circle overlaps with a portion of the Vehicles circle). Statement 3: No truck is a bicycle. (Trucks circle and Bicycles circle are separate). Now, let's analyze the conclusions: I. Some cars are trucks. Cars are inside Vehicles. Trucks overlap with Vehicles. However, the overlap of Trucks and Vehicles might or might not include the Cars part of Vehicles. There's no definite link to conclude 'Some cars are trucks'. So, Conclusion I does not necessarily follow. II. No car is a bicycle. We know 'All cars are vehicles' and 'No truck is a bicycle'. Also, 'Some vehicles are trucks'. This means there are vehicles that are trucks and vehicles that are not trucks. Cars are vehicles. If a car is not a truck, it could potentially be a bicycle, as there is no direct link established between 'Cars' and 'Bicycles' through the premises that applies to all cars. So, Conclusion II does not necessarily follow. Therefore, neither conclusion I nor II follows.

Solution

Correct: A

Let's represent the statements: Statement 1: Some men are wise. (Overlap between Men and Wise circles). Statement 2: No wise person is a fool. (Wise circle and Fool circle are separate). Now, let's analyze the conclusions: I. Some men are not fools. From Statement 1, there is a portion of men who are wise. From Statement 2, no wise person is a fool. Therefore, that specific portion of men (who are wise) cannot be fools. This implies 'Some men are not fools'. Thus, Conclusion I follows. II. All wise persons are men. We only know that 'Some men are wise'. It is possible that there are wise persons who are not men (e.g., women, or other categories). So, Conclusion II does not necessarily follow. Therefore, only conclusion I follows.

Solution

Correct: D

Let's represent the statements: Statement 1: All laptops are computers. (Laptops circle is entirely inside Computers circle). Statement 2: Some computers are expensive. (Expensive circle overlaps with a portion of the Computers circle). Now, let's analyze the conclusions: I. Some laptops are expensive. Laptops are inside Computers. Expensive things overlap with Computers. However, the overlap of Expensive things and Computers might or might not include the Laptops part of Computers. There's no definite link to conclude 'Some laptops are expensive'. So, Conclusion I does not necessarily follow. II. Some expensive things are laptops. This is a conversion of Conclusion I. Since Conclusion I is not necessarily true, its conversion is also not necessarily true. So, Conclusion II does not necessarily follow. Therefore, neither conclusion I nor II follows.

Solution

Correct: A

Let's represent the statements: Statement 1: All roads are streets. (Roads circle is entirely inside Streets circle). Statement 2: Some streets are pathways. (Pathways circle overlaps with a portion of the Streets circle). Statement 3: All pathways are narrow. (Pathways circle is entirely inside Narrow circle). From Statement 2 and 3: 'Some streets are pathways' and 'All pathways are narrow'. This implies 'Some streets are narrow'. Now, let's analyze the conclusions: I. Some narrow are streets. This is a conversion of 'Some streets are narrow', which we derived from the premises. If 'Some A are B', then 'Some B are A' is a valid conversion. Thus, Conclusion I follows. II. Some roads are narrow. Roads are inside Streets. Some Streets are Narrow. However, the part of Streets that are narrow might not include the Roads. There's no definite link to conclude 'Some roads are narrow'. So, Conclusion II does not necessarily follow. Therefore, only conclusion I follows.

Solution

Correct: B

Let's represent the statements: Statement 1: No house is an apartment. (Houses circle and Apartments circle are separate). Statement 2: All apartments are buildings. (Apartments circle is entirely inside Buildings circle). Now, let's analyze the conclusions: I. No house is a building. We know 'No house is an apartment' and 'All apartments are buildings'. This means the part of buildings that are apartments are not houses. However, there could be other buildings (not apartments) that could be houses. So, Conclusion I does not necessarily follow. II. Some buildings are not houses. Since 'All apartments are buildings' (Statement 2) and 'No house is an apartment' (Statement 1), it implies that the group of 'apartments' (which are also 'buildings') cannot be 'houses'. Therefore, 'Some buildings (specifically, the apartments that are buildings) are not houses' is a valid conclusion. Thus, Conclusion II follows. Therefore, only conclusion II follows.

Solution

Correct: E

Let's represent the statements: Statement 1: Some writers are poets. (Overlap between Writers and Poets circles). Statement 2: All poets are artistic. (Poets circle is entirely inside Artistic circle). Combining the statements: Since 'Some writers are poets' and 'All poets are artistic', it logically follows that the portion of writers who are poets must also be artistic. Therefore, 'Some writers are artistic'. Now, let's analyze the conclusions: I. Some writers are artistic. This conclusion directly follows from the combination of the two statements. Thus, Conclusion I follows. II. Some artistic are writers. This is a conversion of 'Some writers are artistic'. If 'Some A are B', then 'Some B are A' is a valid conversion. Thus, Conclusion II follows. Therefore, both conclusions I and II follow.

Solution

Correct: E

Let's represent the statements: Statement 1: All files are folders. (Files circle is entirely inside Folders circle). Statement 2: No folder is a document. (Folders circle and Documents circle are separate). Now, let's analyze the conclusions: I. No file is a document. Since 'All files are folders' and 'No folder is a document', it means that if files are inside folders, and folders are separate from documents, then files must also be separate from documents. Thus, Conclusion I follows directly. II. Some folders are files. If 'All files are folders', it means that all the members of the 'Files' set are also members of the 'Folders' set. Therefore, 'Some folders are files' (specifically, the folders that contain files) is a valid conclusion. Thus, Conclusion II follows. Therefore, both conclusions I and II follow.

Solution

Correct: C

Let's represent the statements: Statement 1: Some metals are hard. (Overlap between Metals and Hard circles). Statement 2: Some hard substances are shiny. (Overlap between Hard and Shiny circles). Now, let's analyze the conclusions: I. Some metals are shiny. We know 'Some metals are hard' and 'Some hard substances are shiny'. However, there is no definite link established between 'metals' and 'shiny'. The hard substances that are metals might not be the same hard substances that are shiny. It's possible for some metals to be shiny, and it's also possible for no metals to be shiny. So, Conclusion I is not necessarily true (uncertain). II. No metal is shiny. Similarly, it's possible that no metals are shiny, but it's not certain. So, Conclusion II is also not necessarily true (uncertain). However, conclusions I (Some metals are shiny) and II (No metal is shiny) form a complementary pair (contradictory pair 'I' type and 'E' type propositions). If 'Some A are B' is false, then 'No A are B' must be true, and vice-versa. Since both are individually uncertain, one of them must be true. Therefore, 'Either conclusion I or II follows'.

Solution

Correct: E

Let's represent the statements: Statement 1: All teachers are educated. (Teachers circle is entirely inside Educated circle). Statement 2: All educated persons are employed. (Educated circle is entirely inside Employed circle). Combining the statements: Since 'All teachers are educated' and 'All educated persons are employed', it logically follows that 'All teachers are employed'. Now, let's analyze the conclusions: I. All teachers are employed. This conclusion directly follows from the combination of the two statements. Thus, Conclusion I follows. II. Some employed persons are teachers. If 'All A are B' is true, then 'Some B are A' is also true. Since 'All teachers are employed' is true, it means that the set of teachers is a subset of employed persons. Therefore, 'Some employed persons are teachers' is also true. Thus, Conclusion II follows. Therefore, both conclusions I and II follow.

Solution

Correct: B

Let's represent the statements: Statement 1: No dog is a cat. (Dogs circle and Cats circle are separate). Statement 2: All cats are furry. (Cats circle is entirely inside Furry circle). Now, let's analyze the conclusions: I. No dog is furry. We know 'No dog is a cat' and 'All cats are furry'. This means dogs are separate from the portion of furry things that are cats. However, there could be other furry things (not cats) that could be dogs. So, Conclusion I does not necessarily follow. II. Some furry things are not dogs. Since 'All cats are furry' (Statement 2) and 'No dog is a cat' (Statement 1), it implies that the group of 'cats' (which are also 'furry') cannot be 'dogs'. Therefore, 'Some furry things (specifically, the cats that are furry) are not dogs' is a valid conclusion. Thus, Conclusion II follows. Therefore, only conclusion II follows.

Solution

Correct: D

Let's represent the statements: Statement 1: Some flowers are red. (Overlap between Flowers and Red circles). Statement 2: Some red things are beautiful. (Overlap between Red and Beautiful circles). Now, let's analyze the conclusions: I. Some flowers are beautiful. We know 'Some flowers are red' and 'Some red things are beautiful'. However, there is no definite link established between 'flowers' and 'beautiful'. The red things that are flowers might not be the same red things that are beautiful. It's possible for some flowers to be beautiful, and it's also possible for no flowers to be beautiful. So, Conclusion I does not necessarily follow. II. Some beautiful things are flowers. This is a conversion of Conclusion I. Since Conclusion I is not necessarily true, its conversion is also not necessarily true. So, Conclusion II does not necessarily follow. Since neither conclusion can be definitively inferred and they do not form an 'either/or' complementary pair (both are 'Some A are B' type statements), neither conclusion I nor II follows.

Solution

Correct: B

Let's represent the statements: Statement 1: All rivers are waters. (Rivers circle is entirely inside Waters circle). Statement 2: Some waters are oceans. (Oceans circle overlaps with a portion of the Waters circle). Statement 3: All oceans are vast. (Oceans circle is entirely inside Vast circle). From Statement 2 and 3: 'Some waters are oceans' and 'All oceans are vast'. This implies 'Some waters are vast'. Now, let's analyze the conclusions: I. Some rivers are vast. Rivers are inside Waters. Some Waters are Vast. However, the part of Waters that are vast might not include the Rivers. There's no definite link to conclude 'Some rivers are vast'. So, Conclusion I does not necessarily follow. II. Some vast things are waters. This is a conversion of 'Some waters are vast', which we derived from the premises. If 'Some A are B', then 'Some B are A' is a valid conversion. Thus, Conclusion II follows. Therefore, only conclusion II follows.

Solution

Correct: E

Let's represent the statements: Statement 1: All books are papers. (Books circle is entirely inside Papers circle). Statement 2: No paper is a pen. (Papers circle and Pens circle are separate). Now, let's analyze the conclusions: I. No book is a pen. Since 'All books are papers' and 'No paper is a pen', it means that if books are inside papers, and papers are separate from pens, then books must also be separate from pens. Thus, Conclusion I follows directly. II. Some papers are books. If 'All books are papers', it means that all the members of the 'Books' set are also members of the 'Papers' set. Therefore, 'Some papers are books' (specifically, the papers that are books) is a valid conclusion. Thus, Conclusion II follows. Therefore, both conclusions I and II follow.

Solution

Correct: A

Let's represent the statements: Statement 1: Some squares are rectangles. (Overlap between Squares and Rectangles circles). Statement 2: No rectangle is a circle. (Rectangles circle and Circles circle are separate). Now, let's analyze the conclusions: I. Some squares are not circles. From Statement 1, there is a portion of squares that are rectangles. From Statement 2, no rectangle is a circle. Therefore, that specific portion of squares (which are rectangles) cannot be circles. This implies 'Some squares are not circles'. Thus, Conclusion I follows. II. All squares are circles. We know 'Some squares are rectangles' and 'No rectangle is a circle'. This implies that at least some squares are not circles. Therefore, 'All squares are circles' cannot be true. So, Conclusion II does not necessarily follow. Therefore, only conclusion I follows.