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:
No chart is a casket.
Not a single casket is a house.
Every house is an adapter.
Conclusions:
(I) Some adapters which are houses are charts as well.
(II) No chart is a house.
(III) Some adapters are houses.
Statements: Chart, Casket, House Relationships
The task is to determine which conclusions logically follow from the given statements. We must assume the statements are true, irrespective of whether they align with common knowledge.
Premises: Chart, Casket, House Logic
Let's break down the relationships described in the premises using set theory notation for clarity:
- Statement 1: No chart is a casket.
Formal Logic: This means the set of Charts (C) and the set of Caskets (K) have no common elements. Represented as: $C \cap K = \emptyset$. - Statement 2: Not a single casket is a house.
Formal Logic: Similarly, the set of Caskets (K) and the set of Houses (H) are disjoint. Represented as: $K \cap H = \emptyset$. - Statement 3: Every house is an adapter.
Formal Logic: This means the entire set of Houses (H) is contained within the set of Adapters (A). Represented as: $H \subseteq A$.
Conclusions: Evaluating Chart, House Links
We will now systematically evaluate each conclusion based on the logical implications of the premises.
Conclusion I: Adapters, Houses, Charts Link
Claim: Some adapters which are houses are charts as well.
Meaning: Since Statement 3 establishes that all Houses are Adapters ($H \subseteq A$), the phrase 'adapters which are houses' simply refers to the set of Houses (H). Therefore, this conclusion claims that there is an overlap between the set of Houses (H) and the set of Charts (C). In set notation: $H \cap C \neq \emptyset$.
Analysis: From Statement 1 ($C \cap K = \emptyset$) and Statement 2 ($K \cap H = \emptyset$), we know that both Charts and Houses are separate from Caskets. However, these statements do not provide any information linking Charts and Houses directly or indirectly. It is possible for them to overlap, or to be completely separate.
Example Scenario: Consider Charts = {1}, Caskets = {2}, Houses = {3}, and Adapters = {3, 4}. Here, Statement 1 ($ \{1\} \cap \{2\} = \emptyset $) and Statement 2 ($ \{2\} \cap \{3\} = \emptyset $) are true. Statement 3 ($ \{3\} \subseteq \{3, 4\} $) is also true. However, Conclusion I ($H \cap C \neq \emptyset$) is false because $ \{3\} \cap \{1\} = \emptyset $. Since we can find a valid case where the conclusion is false, it does not logically follow.
Verdict: Conclusion I does not follow.
Conclusion II: Chart House Disjoint Check
Claim: No chart is a house.
Meaning: This conclusion asserts that the set of Charts (C) and the set of Houses (H) have no members in common. In set notation: $C \cap H = \emptyset$.
Analysis: Just like with Conclusion I, the premises $C \cap K = \emptyset$ and $K \cap H = \emptyset$ do not logically necessitate that $C \cap H = \emptyset$. The separation from Caskets does not imply separation between Charts and Houses.
Example Scenario: Let Charts = {1, 2}, Caskets = {3, 4}, Houses = {1, 5}, and Adapters = {1, 5, 6}. The statements hold: No {1, 2} is {3, 4}. No {3, 4} is {1, 5}. Every {1, 5} is in {1, 5, 6}. However, Conclusion II ($C \cap H = \emptyset$) is false because the element '1' exists in both the Charts set and the Houses set.
Verdict: Conclusion II does not follow.
Conclusion III: Adapter House Overlap Check
Claim: Some adapters are houses.
Meaning: This conclusion states that there is at least one element that belongs to both the set of Adapters (A) and the set of Houses (H). In set notation: $A \cap H \neq \emptyset$.
Analysis: Statement 3 directly states $H \subseteq A$. This means every element in H is also in A. Assuming the set of Houses is not empty (which is a standard assumption in such logic problems unless specified otherwise), there must be at least one house. Because every house is an adapter, this existing house is also an adapter. Therefore, the intersection of Adapters and Houses ($A \cap H$) is precisely the set of Houses (H), which we assume is non-empty. Hence, $A \cap H \neq \emptyset$.
Verdict: Conclusion III follows.
Deduction: Final Conclusion Validity
After analyzing each conclusion against the given statements:
- Conclusion I does not logically follow.
- Conclusion II does not logically follow.
- Conclusion III logically follows.
Therefore, the only conclusion that is logically guaranteed by the premises is Conclusion III.
Similar Questions
- In this question, 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 follows/follow from the statements.
Statements:
I. Some tables are chairs.
II. All desks are chairs.
III. No chair is a fan.
Conclusions:
I. Some desks are tables.
II. Some fans are desks. - In this question, 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 follows/follow from the statements.
Statements:
I. All tables are charts.
II. Some charts are graphs.
III. Some graphs are pies.
Conclusions:
I. No table is a graph.
II. Some pies are charts. - In this question, 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 follows/follow from the statements.
Statements:
All boxes are squares.
All squares are rectangles.
All rectangles are figures.
Conclusions:
I. All boxes are figures.
II. All squares are figures. 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 given conclusions logically follow(s) from the statements.
Statements:
Some fractions are marts.
No mart is a stud.
All studs are frogs.
Conclusions:
I. Some fractions are definitely not studs.
II. Some fractions are definitely not marts.
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 calculators are lanterns.
Some lanterns are mobiles.
Conclusions:
I. All mobiles are calculators.
II. No mobile is a lantern.
III. Some mobiles are lanterns.
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 parrots are crows.
All crows are eagles.
Conclusions:
I. All parrots are eagles.
Il. Some crows are parrots.
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 from the statements.
Statements:
No certificate is a paper.
All pens are papers.
Conclusions:
I. No certificate is a pen.
II. No paper is a certificate.
III. Some papers are pens.
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 officers are clerks.
All clerks are workers.
Conclusions:
I. Some workers are officers.
Il. No worker is an officer.
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 chargers are phones.
No phone is a watch.
Conclusions:
I. All phones are chargers.
Il. No charger is a watch.
Ill. No watch is a phone.
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 wallets are bags.
Some bags are envelopes.
Conclusions:
I. Some wallets are envelopes.
II. No wallet is an envelope.
III. No bag is an envelope.
Important Questions from 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:
Some students are players.
All players are male.
Conclusions:
I. All males are players.
II. Some males are students.
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:
No creative is an employer.
All experts are creative.
All workers are experts.
Conclusions:
I. No employer is an expert.
II. No worker is an employer.
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 actors are choreographers.
All choreographers are producers.
Not a single producer is a director.
Conclusions:
I. Some actors are directors.
II. Not a single actor is a director.
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 flowers are tulips.
2. No tulips are whites.
Conclusions :
I. All tulips are flowers.
II. Some tulips are whites.
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 rats are dogs.
Some rats are hens.
Conclusions:
I. Some rats are dogs.
II. Some hens are rats.
