Introduction / Context:
This problem asks which statement is logically entailed by three facts about sets of dogs and their properties (running, swimming, resembling masters). We must accept the facts as universally and existentially quantified exactly as stated and avoid adding extra assumptions.
Given Data / Assumptions:
- All dogs like to run (every dog is in the “runs” set).
- Some dogs like to swim (there exists at least one swimmer dog).
- Some dogs look like their masters (there exists at least one look-alike dog).
- Sets may overlap in any way not prohibited by the facts.
Concept / Approach:
- Translate each candidate statement and test against the facts. A statement “must be true” only if it is unavoidable given the facts.
- Remember: “All dogs like to run” implies any subset of dogs (including swimmers) like to run.
Step-by-Step Solution:
Evaluate I: “All dogs who like to swim look like their masters.” Not required by the facts; swimmers may or may not look like their masters. So I need not be true.Evaluate II: “Dogs who like to swim also like to run.” Since all dogs like to run, every swimmer (being a dog) likes to run. II must be true.Evaluate III: “Dogs who like to run do not look like their masters.” Since some dogs look like their masters and all dogs like to run, at least one runner does look like its master. So III is false.
Verification / Alternative check:
Construct a Venn diagram mentally: the “runs” circle contains all dogs. “swims” and “look-like” are arbitrary subsets. II is forced; I and III are not.
Why Other Options Are Wrong:
I only: Incorrect because I is not entailed.II and III only: III is contradicted by the existence of look-alike dogs within the universal runners.None of the statements is a known fact: Wrong because II is guaranteed.
Common Pitfalls:
Confusing “some” with “all,” or overlooking that universal facts propagate to every subset.
Final Answer:
II only
Discussion & Comments