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: 1. All rugs are blankets. 2. All blankets are pillows. 3. Some blankets are frames. Conclusions: I. All pillows are rugs. II. Some pillows are rugs. III. All rugs are frames
Only conclusion II follows
Let's analyze the given statements and conclusions based on the principles of logical deduction. This type of problem, involving categorical statements like "All A are B" and "Some A are B," falls under the category of syllogism or deductive reasoning.
Analyzing the Given Statements on Rugs, Blankets, and Pillows
We are given the following statements:
- All rugs are blankets.
- All blankets are pillows.
- Some blankets are frames.
We need to assume these statements are true, even if they contradict common knowledge. Our goal is to determine which of the given conclusions logically follow from these statements.
Combining Statements for Logical Inference
Let's combine the first two statements:
- Statement 1: All rugs are blankets. (If something is a rug, it must be a blanket.)
- Statement 2: All blankets are pillows. (If something is a blanket, it must be a pillow.)
From these two statements, we can logically deduce a relationship between rugs and pillows. If all rugs are blankets, and all those blankets are also pillows, then it must be true that all rugs are pillows. This is a transitive property in logic.
So, a key deduction is: All rugs are pillows.
Statement 3 tells us that some blankets are frames. This establishes a relationship between blankets and frames, but it doesn't directly link rugs or pillows to frames based on "all" or "none" type relationships, only "some".
Evaluating Each Conclusion Based on the Statements
Now, let's examine each conclusion:
Conclusion I: All pillows are rugs.
We deduced that "All rugs are pillows". Conclusion I states the reverse: "All pillows are rugs". Let's consider if this must be true based on the statements.
Statements 1 and 2 tell us that the set of rugs is entirely contained within the set of blankets, and the set of blankets is entirely contained within the set of pillows. This means the set of rugs is a subset of the set of pillows.
Consider an example: If all apples are fruits, it doesn't mean all fruits are apples (a banana is a fruit but not an apple). Similarly, if all rugs are pillows, it doesn't necessarily mean all pillows are rugs. There could be pillows that are blankets but not rugs, or even pillows that are not blankets at all (although statement 2 says all blankets are pillows, it doesn't restrict pillows from existing outside of blankets).
Therefore, the statement "All pillows are rugs" does not logically follow from the given information.
Conclusion II: Some pillows are rugs.
We deduced that "All rugs are pillows". If it is true that all members of set A are also members of set B, then it must also be true that some members of set B are members of set A (assuming set A is not empty, which is typically the assumption in these problems unless stated otherwise). If there is at least one rug, and every rug is also a pillow, then there must be at least one pillow that is a rug. So, some pillows are indeed rugs.
Therefore, the statement "Some pillows are rugs" logically follows from the deduction that all rugs are pillows.
Conclusion III: All rugs are frames.
We know "All rugs are blankets" and "Some blankets are frames". This means the set of rugs is inside the set of blankets, and there is an overlap between the set of blankets and the set of frames.
Consider this scenario: Suppose blankets A, B, C, D exist. Rugs are A and B. Blankets C and D are frames. Blankets A and B are not frames. This scenario is consistent with "All rugs (A, B) are blankets (A, B, C, D)" and "Some blankets (C, D) are frames". In this scenario, rugs (A, B) are not frames. Thus, "All rugs are frames" is not necessarily true.
It is possible that the "some blankets" that are frames happen to include all the rugs, but it is not guaranteed by the statements.
Therefore, the statement "All rugs are frames" does not logically follow from the given information.
Summary of Conclusions
- Conclusion I (All pillows are rugs): Does not follow.
- Conclusion II (Some pillows are rugs): Follows.
- Conclusion III (All rugs are frames): Does not follow.
Based on this analysis, only conclusion II logically follows from the given statements.
Final Answer
Based on the detailed analysis of each conclusion, only Conclusion II is logically derived from the provided statements.
| Conclusion | Analysis | Logically Follows? |
|---|---|---|
| I. All pillows are rugs. | Converse of "All rugs are pillows". Not necessarily true. | No |
| II. Some pillows are rugs. | Directly follows from "All rugs are pillows". | Yes |
| III. All rugs are frames. | Relationship between rugs and frames is not guaranteed by the statements. | No |
Revision Table: Statements and Conclusions Logic
Understanding the relationship between different categories is key in these logic problems. Here’s a quick recap of the deductions made:
- Statement 1: All Rugs $\subseteq$ Blankets
- Statement 2: All Blankets $\subseteq$ Pillows
- Deduction: All Rugs $\subseteq$ Pillows
- Statement 3: Some Blankets $\cap$ Frames $\neq \emptyset$
Using these relationships, we checked if the conclusions are necessarily true.
Additional Information: Types of Logical Statements
Statements used in syllogisms typically fall into one of four types:
- Universal Affirmative (A): All S are P (e.g., All rugs are blankets).
- Universal Negative (E): No S are P (e.g., No rugs are frames).
- Particular Affirmative (I): Some S are P (e.g., Some blankets are frames).
- Particular Negative (O): Some S are not P (e.g., Some blankets are not rugs).
Understanding these types helps in drawing diagrams or using rules to check the validity of conclusions.
Similar Questions
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. All T are G.
II. All G are H.
Conclusions:
I. All T are H.
II. No G is T.
III. All H are G.
Three statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All reds are pinks.
All whites are pinks.
All pinks are oranges.
Conclusions:
I. All whites are oranges.
II. All oranges are reds.
III. Some oranges are whites.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. All R are M.
II. All L are M.
Conclusions:
I. Some M are not R.
II. Some L are R.
III. Some M are L.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. All M are A.
II. No A is R.
Conclusion:
I. No M is R.
II. Some A are M.
III. Some R are A.
Three statements are given, followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All c are f.
No f is t.
All p are c.
Conclusions:
I. Some t are c.
II. No p is t.
Three Statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
Some bottles are towels.
No towel is a pillow.
All bottles are coats.
Conclusions:
I. Some coats are towels.
II. No coat is a towel.
III. Some bottles are pillows.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:
All Tim are Jack.
No Jack is Ben.
Some Ben is Ruth.
Conclusions:
I. No Ruth is Jack.
II. Some Ben are Jack.
III. No Ben is Tim.
In this question, three statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follows/follow from the statements.
Statements:
Some roses are yellow.
Some yellow are buses.
All buses are green.
Conclusions:
I. All yellow are roses.
II. Some buses are yellow.
III. All yellow are green.
- Sunita travels 6 Km towards north east. Then travels 9 Km towards west. From there, she goes 6 Km south-west and finally goes 3 Km towards east. How far is she from her initial position?
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:
Some laptops are phones.
Some phones are capsules.
Some capsules are pens.
Conclusions:
I. Some laptops are capsules.
II. Some phones are pens.
III. Some pens are capsules.
IV. Some pens are phones.
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.
