In this categorical-logic question, treat ‘rectangle’ and ‘square’ as…
2024
In this categorical-logic question, treat ‘rectangle’ and ‘square’ as arbitrary category labels; assume no geometric relationship beyond the given statement. If ‘Some rectangles are squares’ is true, which group lists exactly the statements that must be false?
Answer: B. No rectangles are squares — ConceptIn categorical logic, “Some A are B” means the intersection of sets A and B is non-empty. It contradicts “No A are B,” but by itself it says nothing…
- A.
Some rectangles are not squares
Some squares are not rectangles
No rectangles are squares
- B.
No rectangles are squares
- C.
Some rectangles are not squares
Some squares are not rectangles
- D.
All rectangles are squares
Some rectangles are not squares
Some squares are not rectangles
No rectangles are squares
Show answer & explanation
Correct answer: B
Concept
In categorical logic, “Some A are B” means the intersection of sets A and B is non-empty. It contradicts “No A are B,” but by itself it says nothing about whether either set has elements outside the intersection or whether one set is wholly contained in the other.
Application
Let R be rectangles and S be squares. The premise gives at least one element in the intersection of R and S.
“All rectangles are squares” is compatible with the premise, so it is not forced false.
“Some rectangles are not squares” and “Some squares are not rectangles” depend on the regions outside the intersection; the premise leaves both regions undetermined.
“No rectangles are squares” requires the intersection to be empty, directly contradicting the given non-empty intersection.
Cross-check
In a Venn diagram, place one element in the overlap. The two outside regions may be empty or non-empty, but the overlap cannot be empty.
Result
Therefore, the only statement that must be false is “No rectangles are squares”; the correct choice is the group containing only that statement.