Read the given statements and conclusions carefully. You have to take the given statements to be true even if they seem to be at variance from commonly known facts. You have to decide which conclusion/s logically and definitely follow/s from the given statements. Statements: Some Salts are Baths. All Tubs are Drums. Some Baths are Tubs. Conclusions: (I) Some Salts are Tubs. (II) Some Drums are Baths.
Only conclusion (II) follows.
Understanding the Logic Problem
This question is a classic example of a syllogism problem, which tests your ability to deduce conclusions based on given statements. In these types of questions, you must assume the statements are true, regardless of whether they align with real-world facts, and determine which conclusions logically and definitely follow.
Analyzing the Given Statements
We are given three statements:
- Some Salts are Baths.
- All Tubs are Drums.
- Some Baths are Tubs.
To solve this, we can use methods like Venn diagrams or rules of syllogism. Venn diagrams are often helpful for visualizing the relationships described by the statements.
Visualizing with Venn Diagrams
Let's represent each category (Salts, Baths, Tubs, Drums) with circles. The statements describe the relationships between these circles.
- Statement 1: Some Salts are Baths. This means there is an overlap between the circle representing Salts (S) and the circle representing Baths (B). We can mark the intersecting area to indicate it is non-empty.
- Statement 3: Some Baths are Tubs. This means there is an overlap between the circle representing Baths (B) and the circle representing Tubs (T). Again, the intersection is non-empty.
- Statement 2: All Tubs are Drums. This is a different type of relationship. It means the entire circle representing Tubs (T) must be completely contained within the circle representing Drums (D). In terms of sets, T $\subset$ D.
Now let's try to combine these relationships:
| Statement | Relationship | Diagrammatic Representation |
|---|---|---|
| Some Salts are Baths | S $\cap$ B $\neq \emptyset$ | Overlapping S and B circles, shaded intersection. |
| Some Baths are Tubs | B $\cap$ T $\neq \emptyset$ | Overlapping B and T circles, shaded intersection. |
| All Tubs are Drums | T $\subset$ D | Circle T drawn completely inside circle D. |
When we combine these, the critical connections are made through 'Baths'. We know some Baths are Tubs (Statement 3). We also know all Tubs are Drums (Statement 2). If a part of Baths is Tubs, and all Tubs are inside Drums, then that specific part of Baths must also be inside Drums.
Evaluating the Conclusions
Now let's examine each conclusion based on our combined understanding from the statements.
Conclusion (I): Some Salts are Tubs.
- We know Some Salts are Baths (S $\cap$ B $\neq \emptyset$).
- We know Some Baths are Tubs (B $\cap$ T $\neq \emptyset$).
Does the overlap between S and B necessarily overlap with T? Not necessarily. Consider the portion of Baths that overlaps with Salts. This portion of Baths could be the part of Baths that is *not* Tubs. Therefore, we cannot definitively say that Some Salts are Tubs based on the given statements. There is no guaranteed direct overlap between Salts and Tubs.
| Statement | Relates |
|---|---|
| Some Salts are Baths | Salts and Baths |
| Some Baths are Tubs | Baths and Tubs |
While there's a link through Baths, it's a "Some... Some..." link, which typically does not guarantee a "Some" conclusion between the outer terms (Salts and Tubs) in this structure.
Therefore, Conclusion (I) does not logically and definitely follow.
Conclusion (II): Some Drums are Baths.
- We know Some Baths are Tubs (B $\cap$ T $\neq \emptyset$). This means there exists at least one entity that is both a Bath and a Tub.
- We know All Tubs are Drums (T $\subset$ D). This means every single Tub is also a Drum.
Let's take that entity which is both a Bath and a Tub (from statement 3). Since this entity is a Tub, and all Tubs are Drums (statement 2), this entity must also be a Drum. Therefore, there exists at least one entity that is both a Bath and a Drum. This is exactly what "Some Baths are Drums" means.
The statement "Some Baths are Drums" is logically equivalent to "Some Drums are Baths".
Thus, Conclusion (II) logically and definitely follows from the given statements.
Summary of Conclusions
- Conclusion (I): Some Salts are Tubs. - Does NOT follow.
- Conclusion (II): Some Drums are Baths. - FOLLOWS.
Based on our analysis, only conclusion (II) follows from the given statements.
Identifying the Correct Option
We determined that only Conclusion (II) logically follows. Let's look at the given options:
- Both conclusions (I) and (II) follow. - Incorrect.
- Both conclusions (I) and (II) do not follow. - Incorrect.
- Only conclusion (I) follows. - Incorrect.
- Only conclusion (II) follows. - Correct.
The option that states only conclusion (II) follows is the correct one.
Revision Table: Key Concepts
| Concept | Explanation | Syllogism Type Examples |
|---|---|---|
| "All A are B" | Every member of set A is also a member of set B. (A $\subset$ B) | All cats are animals. |
| "Some A are B" | At least one member of set A is also a member of set B. (A $\cap$ B $\neq \emptyset$) | Some students are athletes. |
| "No A are B" | No member of set A is a member of set B. (A $\cap$ B $= \emptyset$) | No dogs are birds. |
| "Some A are not B" | At least one member of set A is not a member of set B. | Some fruits are not sweet. |
Additional Information: Syllogism Rules
Syllogism problems often involve combining two statements (premises) to derive a third statement (conclusion). There are formal rules for what conclusions can be drawn based on the types of premises. For instance:
- If you have "All A are B" and "All B are C", you can conclude "All A are C".
- If you have "Some A are B" and "All B are C", you can conclude "Some A are C". (This is similar to the structure leading to our conclusion II: Some Baths are Tubs, All Tubs are Drums $\rightarrow$ Some Baths are Drums).
- If you have "Some A are B" and "Some B are C", you generally cannot draw a definite conclusion about the relationship between A and C. (This is relevant to why our conclusion I did not follow: Some Salts are Baths, Some Baths are Tubs $\rightarrow$ No definite conclusion about Salts and Tubs).
Understanding these basic structures and how to represent them with Venn diagrams or logical notation is key to solving syllogism questions accurately.
Important Questions from Conventional Syllogism
In this question, three statements followed by two conclusions numbered I and II have been given. You have to take the given statements to be true even if they seem to be at variance from the commonly facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statements: Some flats are apartments.
No apartment is a hall.
Some halls are rooms.
Conclusions: I. At least some rooms are flats.
II. No apartment is a room.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
I. Some blue are red.
II. Some green are red.
Conclusions:
I. No blue is green.
II. No red is green.
Given below are two statements, Consider these statements to be true even if they seem factuality absurd, Read the conclusions and then decide which of the given conclusions logically follow(s) from the given statements?
Statements :
1. All vases are flowers.
2. No flowers is a plant.
Conclusions :
1. No vases is a plant.
2. Some plant are vases
In the question two statements are given, followed by three conclusions, I, II and III. You have to consider the statements to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements.
Statement 1 : Some cars are scooters.
Statement 2 : All scooters are buses.
Conclusion I : Some scooters are cars.
Conclusion II : Some buses are cars.
Conclusion III : All cars are buses.
In the question below are given three statements followed by two conclusions. You have to take each of the given statements to be true. Read the conclusions and then decide which of the conclusions can be logically derived.
Statements:
Some schools are colleges.
No college is university.
All universities are hospitals.
Conclusions:
I. Some schools are universities.
II. At least some hospitals are colleges.
