UGCNET Questions

The first statement says that all rational numbers are contained within the set of real numbers. The second statement tells us that there is at least some overlap between real numbers and integers. From these premises we can be sure that the set of rationals lies inside the reals, but we are not given an explicit link between rationals and integers. It is therefore safe to say that some real numbers may be rational numbers, since rational numbers form a subset of the reals.

Option A:
Option A, “Some integers are rational numbers,” is true in standard mathematics but does not follow solely from the given premises, which never state a direct relation between integers and rationals.

Option B:
Option B, “Some rational numbers are integers,” is also mathematically true but again goes beyond the information explicitly stated in the premises, which only connect rationals with reals and integers with reals.

Option C:
Option C cautiously asserts a possibility: since all rationals are reals, it is entirely consistent that some real numbers may be rational. This conclusion remains within what is warranted by the premises.

Option D:
Option D, “All integers are rational numbers,” is stronger than anything that can be deduced from the given statements and is not logically enforced by them, even though it is true in standard number systems.

In traditional logic, the four standard forms of categorical propositions are labelled A, E, I and O. “Some S are P” is the pattern of a particular affirmative, symbolised as an I proposition. It asserts that at least one member of the subject class belongs to the predicate class. The statement about some scientists being philosophers fits exactly into this structure.

Option A:
Option A, universal affirmative, corresponds to “All S are P,” which would be “All scientists are philosophers,” a much stronger claim than “some.”

Option B:
Option B, universal negative, would say “No scientists are philosophers,” which directly contradicts the given statement.

Option C:
Option C matches the “some… are…” form, making it a particular affirmative that asserts existence of at least one overlapping case.

Option D:
Option D, particular negative, would say “Some scientists are not philosophers,” which has different logical content than the original statement.

The statement “Some students are diligent” is an existential claim asserting that at least one member of the class of students has the property of being diligent. Its contradictory is “No student is diligent.” Therefore, if “some students are diligent” is true, then the statement “no student is diligent” must be false. Saying “It is not the case that no student is diligent” is logically equivalent to affirming that at least one diligent student exists.

Option A:
Option A is stronger than “some”; it claims that all students are diligent, which need not follow from the existence of at least one diligent student.

Option B:
Option B is the direct negation of the original statement and cannot be true if “some students are diligent” is true.

Option C:
Option C correctly denies the universal negative and is consistent with the existential affirmation about diligent students.

Option D:
Option D narrows the claim to exactly one student, which again is stronger than “some” and does not logically follow.