Statements: Some boys are dancers. Some dancers are singers. All singers are…
2023
Statements: Some boys are dancers.
Some dancers are singers.
All singers are directors.
Conclusions: I. Some directors are boys. II. All dancers are directors. Assuming that the information given in the statements is true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follows/follow from the statements.
- A.
Only conclusion I follows
- B.
Only conclusion II follows
- C.
Neither conclusion I nor II follows
- D.
Both conclusion I and II follow
Attempted by 252 students.
Show answer & explanation
Correct answer: C
Final Answer: Neither conclusion I nor II follows.
Given statements:
Some boys are dancers.
Some dancers are singers.
All singers are directors.
Analysis:
You cannot assume transitivity for "some" statements. From "some boys are dancers" and "some dancers are singers" it does not follow that some boys are singers; the particular dancers referred to may be different groups.
Because all singers are directors, only those dancers who are singers are guaranteed to be directors. Since only some dancers are singers, you cannot conclude that all dancers are directors.
Concrete counterexample that satisfies all statements but makes both conclusions false:
Let Boys = {b}, Dancers = {b, s}, Singers = {s}, Directors = {s}.
Then "Some boys are dancers" holds because b is both boy and dancer.
"Some dancers are singers" holds because s is both dancer and singer.
All singers are directors holds because s is a singer and a director.
In this example:
No boy is a director (b is not in Directors), so conclusion I (some directors are boys) is false in this model.
Not all dancers are directors (b is a dancer but not a director), so conclusion II (all dancers are directors) is false in this model.
Since a logically consistent example exists where both conclusions fail while the given statements remain true, neither conclusion follows from the statements.