Direction: In the question below are given some statements followed by some conclusions. You have to take given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts. Statements: All Machines are Transformers All Transformers are Motors Only a few Machines are Generators Conclusions: I. Some Generators are Transformers
Both conclusions I and II follow
Logical Reasoning: Syllogism Analysis
This problem involves analyzing logical statements (syllogisms) to determine which conclusions necessarily follow from the premises, even if they contradict common knowledge.
Understanding the Statements
- Statement 1:
All Machines are Transformers. This establishes a complete inclusion relationship. If something is a Machine, it must also be a Transformer. Mathematically, we can denote this as: $Machines \subseteq Transformers$. - Statement 2:
All Transformers are Motors. Similar to the first statement, this means the set of Transformers is entirely contained within the set of Motors. Represented as: $Transformers \subseteq Motors$. - Statement 3:
Only a few Machines are Generators. This is a crucial statement indicating a partial relationship. It means:- At least one Machine is a Generator ($Some\ Machines \cap Generators \neq \emptyset$).
- At least one Machine is NOT a Generator ($Some\ Machines \cap (NOT\ Generators) \neq \emptyset$).
Analyzing the Conclusions
Conclusion I: Some Generators are Transformers
To evaluate this, we combine Statement 1 and Statement 3:
- From Statement 3, we know that Some Machines are Generators.
- From Statement 1, we know that All Machines are Transformers.
Since the group of Machines that are Generators belongs to the set of Machines, and every Machine is also a Transformer, it logically follows that the Machines which are Generators must also be Transformers. Therefore, Some Generators are Transformers is a valid conclusion.
Using notation: ($Some\ M\ are\ G$) and ($All\ M\ are\ T$). Combining these implies that the intersection of G and M is a subset of T. Hence, ($Some\ G\ are\ T$).
Conclusion II: Some Motors are Machines
Here, we combine Statement 1 and Statement 2:
- Statement 1:
All Machines are Transformers($Machines \subseteq Transformers$). - Statement 2:
All Transformers are Motors($Transformers \subseteq Motors$).
By linking these two statements (a transitive property), we can deduce that All Machines are Motors ($Machines \subseteq Motors$). If every single Machine is a Motor, it naturally means that some part of the Motors set must be Machines. Therefore, Some Motors are Machines is also a valid conclusion.
Using notation: ($All\ M\ are\ T$) + ($All\ T\ are\ Mo$) $\implies$ ($All\ M\ are\ Mo$). If all M are Mo, then ($Some\ Mo\ are\ M$) must be true.
Final Verdict
Based on the analysis, both Conclusion I and Conclusion II logically and necessarily follow from the given statements.
Similar Questions
- Direction: In the question below are given three statements followed by two conclusions numbered I, II. You have to take the given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements disregarding commonly known fact.Statements:All A are B.Some B are C.Some C are D.Conclusions:l. All B are A.ll. All D are C.
Direction: In the question below are given three statements followed by two conclusions. You have to take the given statements to be true even if they seem to be at variance from commonly known facts. Read all the given conclusions and then decide which of the following conclusions logically follows from the given statements disregarding commonly known facts.
Statements:
Some sketch are not creative.
All creative are pen.
A few sketch are ink.
Conclusions:
I. Some pen are ink.
II. No pen is ink.
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:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
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 directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
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 follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
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 cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
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 employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
