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
The statements below are followed by conclusions labeled I, II and III. 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:
All twenty are thirty.
All thirty are forty.
All forty are sixty.
All sixty are seventy.
Conclusions:
I. Some forty are thirty.
II. Some seventy are sixty.
III. No thirty is twenty.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 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:
All Strong are animals.
Some animals are Tigers.
All Tigers are Sharp.
Conclusions:
I. Some Strong are Sharp.
II. No Strong is Sharp.The statements below are followed by conclusions labelled I, II and III. 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 women are weak.
Some weaks are female.
All female are iron.
All iron are gold.Conclusions:
I. Some weaks are iron.
II. Some gold are weaks.
III. Some women are female.
The 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 teaspoons are glasses.
All teddies are teaspoons.
Conclusions:
I. Some teddies are glasses.
II. Some glasses are teddies.
The 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:
All bangles are rings.
Some rings are toys.
Some toys are dolls.
Conclusions:
I. Some rings are bangles.
II. Some dolls are rings.
