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(s) logically follows the given statements.
Statements:
I. Some yellow are green.
II. All white are green.
Conclusions:
I. Some green are yellow.
II. Some green are white.
Statements Logic: Yellow, Green, White
This question involves logical reasoning, specifically analyzing syllogism statements and conclusions. We must assume the given statements are true, even if they seem unusual, and determine if the conclusions necessarily follow from them.
Statement I: Some yellow are green.
This statement indicates that there is at least one instance where something is both 'yellow' and 'green'. The categories are not mutually exclusive; they overlap.
In terms of sets, if Y represents the set of yellow things and G represents the set of green things, this statement means their intersection is non-empty: $Y \cap G \neq \emptyset$.
Statement II: All white are green.
This statement means that every member of the category 'white' is also a member of the category 'green'. The 'white' category is entirely contained within the 'green' category.
Using set notation, if W represents the set of white things, this statement means W is a subset of G: $W \subseteq G$.
Conclusions Validity Check
Conclusion I: Some green are yellow.
Statement I establishes that "Some yellow are green". This implies that the overlap between yellow and green is mutual. If there exists at least one thing that is both yellow and green ($Y \cap G \neq \emptyset$), then it must also be true that there exists at least one thing that is both green and yellow ($G \cap Y \neq \emptyset$).
Therefore, Conclusion I, "Some green are yellow", logically follows from Statement I.
Conclusion II: Some green are white.
Statement II establishes that "All white are green" ($W \subseteq G$). Since the set of white things is entirely contained within the set of green things, it means that any white thing is also a green thing. This directly implies that there are indeed some green things that are white. Assuming the category 'white' is not empty, the intersection $G \cap W$ is non-empty.
Therefore, Conclusion II, "Some green are white", logically follows from Statement II.
Final Logical Determination
We have determined that:
- Conclusion I logically follows from Statement I.
- Conclusion II logically follows from Statement II.
Because both conclusions are valid deductions based on the provided statements, the correct answer must indicate that both conclusions follow.
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.
