Read the given statements and conclusions carefully. Assuming that the information given in the statement 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 fans are coolers. All coolers are equipment. Some equipment are mobiles. Conclusions: I. Some equipments are fans. II. Some mobiles are fans. III. No equipment is fan and cooler both
Only conclusion I follows
Solving Logical Syllogism Problems
This question asks us to analyze given statements and determine which of the provided conclusions logically follow. This type of problem is known as a syllogism in logical reasoning. We need to assume the statements are true, even if they contradict common knowledge, and use them as the sole basis for evaluating the conclusions.
Let's break down the given information:
Statements Analysis
- Statement 1: Some fans are coolers.
This indicates an overlap between the categories 'fans' and 'coolers'. There is at least one entity that belongs to both groups.
- Statement 2: All coolers are equipment.
This is a universal affirmative statement. It means that the set of 'coolers' is completely contained within the set of 'equipment'. If something is a cooler, it must also be equipment.
- Statement 3: Some equipment are mobiles.
This indicates an overlap between the categories 'equipment' and 'mobiles'. There is at least one entity that belongs to both groups.
Conclusions Evaluation
Now, let's evaluate each conclusion based on the statements:
Conclusion I: Some equipments are fans.
Let's combine statements 1 and 2:
- Some fans are coolers (Statement 1)
- All coolers are equipment (Statement 2)
Since some fans are coolers, and all coolers are equipment, it means that the fans that are coolers must also be equipment. Therefore, there are some fans that are equipment. This implies that some equipment are fans.
Consider a Venn diagram approach: Draw circles for Fans, Coolers, and Equipment. Statement 1 shows an overlap between Fans and Coolers. Statement 2 shows the entire Coolers circle inside the Equipment circle. The overlap region between Fans and Coolers is within the Coolers circle, which is itself within the Equipment circle. Thus, the overlap region between Fans and Coolers is also within the Equipment circle. This overlap region represents entities that are Fans and Coolers and also Equipment. Since this region is not empty (from Statement 1), it means there are some entities that are Fans and Equipment. So, some Equipment are Fans.
Conclusion I logically follows from the statements.
Conclusion II: Some mobiles are fans.
Let's look at the statements involving mobiles and fans:
- Some fans are coolers (Statement 1)
- All coolers are equipment (Statement 2)
- Some equipment are mobiles (Statement 3)
From Statements 1 and 2, we established that Some fans are equipment. Statement 3 says Some equipment are mobiles. We have "Some F are E" and "Some E are M". Can we conclude "Some F are M"? Not necessarily.
Using Venn diagrams: We have overlap between F and C (which is inside E), and overlap between E and M. The overlap between E and M could be entirely within the part of E that does NOT overlap with F. There is no direct or guaranteed connection established between Fans and Mobiles through these statements. It's possible that the equipment that are mobiles are not the same equipment that are fans.
Conclusion II does not logically follow from the statements.
Conclusion III: No equipment is fan and cooler both.
Let's re-examine statements 1 and 2:
- Some fans are coolers (Statement 1)
- All coolers are equipment (Statement 2)
Statement 1 explicitly states that "Some fans are coolers". This means there is an intersection between the set of fans and the set of coolers. Let's call this intersection $F \cap C$. This intersection is not empty; it contains at least one element.
Statement 2 says "All coolers are equipment". This means the set of coolers ($C$) is a subset of the set of equipment ($E$), i.e., $C \subseteq E$.
If an entity is in the intersection $F \cap C$, it is both a fan and a cooler. Since all coolers are equipment, any entity in $F \cap C$ must also be in $E$. Therefore, the intersection $F \cap C$ is also a subset of $E$. This means there are entities that are both fans and coolers AND equipment.
Conclusion III says "No equipment is fan and cooler both". This is equivalent to saying that the set of equipment that are also fans and coolers is empty. In other words, $E \cap (F \cap C)$ is empty. However, as we've shown, if $F \cap C$ is not empty and $C \subseteq E$, then $F \cap C$ must be a subset of $E$, meaning $E \cap (F \cap C) = F \cap C$, which is not empty.
Thus, Conclusion III directly contradicts the information derived from statements 1 and 2.
Conclusion III does not logically follow from the statements; it is false based on the statements.
Summary of Conclusions
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I. Some equipments are fans. | Yes | Derived from Some F are C and All C are E $\implies$ Some F are E $\implies$ Some E are F. |
| II. Some mobiles are fans. | No | No direct or guaranteed link between Fans and Mobiles established by the statements. |
| III. No equipment is fan and cooler both. | No | Statements imply that there are entities that are Fan AND Cooler AND Equipment. Conclusion III contradicts this. |
Based on our analysis, only Conclusion I logically follows from the given statements.
Revision Table: Key Syllogism Rules
| Type of Statement | Relationship | Example |
|---|---|---|
| All A are B (A-type) | A is a subset of B | All dogs are mammals. |
| No A are B (E-type) | A and B are disjoint sets | No cats are dogs. |
| Some A are B (I-type) | A and B have at least one element in common (overlap) | Some students are engineers. |
| Some A are not B (O-type) | At least one element of A is not in B | Some birds are not eagles. |
Additional Information on Syllogism Inference
Syllogisms involve drawing a conclusion from two premises (statements). The validity of a syllogism depends on its form, not the truth of the premises in the real world. We often use Venn diagrams or formal rules to test validity.
Key points to remember when solving syllogism problems:
- Always assume the statements are true.
- Do not use outside knowledge.
- A conclusion follows only if it MUST be true whenever the premises are true. If there is any possibility, however remote, that the conclusion is false while the premises are true, the conclusion does not follow.
- Statements with "Some" imply existence (at least one).
- Statements with "All" are universal.
In this specific problem, the combination of "Some A are B" and "All B are C" is a classic pattern that leads to the conclusion "Some A are C". This is what we saw with "Some fans are coolers" and "All coolers are equipment", leading to "Some fans are equipment" (or Some equipment are fans, which is equivalent).
Similar Questions
Three Statements are given followed by Four conclusions numbered I, II, III and IV. 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 curtains are pillows.
No pillow is a bedsheet.
Some curtains are bedsheets.
Conclusions:
I. No curtain is a bedsheet.
II. No pillow is a curtain.
III. Some bedsheets are curtains.
IV. Some pillows are curtains.
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 dates are cashews.
All cashews are raisins.
Some raisins are almonds.
Conclusions:
I. All almonds being dates is a possibility.
II. All dates are raisins.
III. No cashew is an almond.
Two 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:
Some vehicles are helicopters.
All helicopters are cars.
Conclusions:
I. All vehicles are cars.
II. Some vehicles are cars.
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 plants are flowers.
Some flowers are fruits.
All flowers are gardens.
Conclusions:
I. Some gardens are plants.
II. Some fruits are gardens.
III. Some gardens are flowers.
Two 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 teachers are engineers.
All clerks are teachers.
Conclusions:
I. Some engineers are clerks.
II. Some teachers are clerks.
III. Some clerks are not engineers.
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 classrooms are schools.
No school is a park.
All houses are parks.
Conclusions:
I. No house is a classroom.
II. No park is a classroom.
III. No school is a house.
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 fans are hammers.
All hammers are fishes.
Conclusions:
I. No fans are fishes.
II. Some fans are fishes.
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 inkpots are kettles.
All utensils are candies.
No kettle is utensil.
Conclusions:
1. Some inkpots are candies.
2. Some kettles are inkpots.
3. Some candies are inkpots.
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 holders are hangers.
All hangers are hammers.
Some hammers are axes.
Conclusions:
I. Some axes are hangers.
II. Some hammers are holders.
III. Some hangers are holders.
Two 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:
Some blankets are quilts.
Some quilts are bedsheets.
Conclusions:
I. All quilts are blankets.
II. All blankets are bedsheets.
Important Questions from Conventional Syllogism
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. Some flowers are chemicals.
2. Some chemicals are herbs.
Conclusions:
I. No flower is a herb.
II. Some flowers are herbs.
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 students are teachers.
All teachers are professors.
Conclusions:
1. All students are professors.
2. All professors are students.
3. No teacher is a professor.
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 dentists are surgeons.
II. All the surgeons are doctors.
Conclusions:
I. Some dentists are doctors.
II. No surgeons are dentists.
The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears at variance with generally established facts. decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
1) Some angers are griefs.
2) Some griefs are loves.
Conclusion:
I. Some griefs are angers.
II. Some loves are griefsThe statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts. decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
Some cows are deer.
Some deer are fishes.
Conclusion:
I. Some cows are fishes.
II. Some fishes are cows.
