Mathematical and Logical Reasoning

Given sentence: "8 is less than 6."

Concept: A sentence is a mathematically acceptable statement if it is either true or false (but not both).

Working: The sentence "8 is less than 6" is false because $8 > 6$. Since it has a definite truth value (false), it qualifies as a statement.

Conclusion: Yes, it is a statement (a false statement).

Given sentence: "Every set is a finite set."

Concept: A sentence is a statement if it is either true or false.

Working: The sentence is false because there exist sets that are not finite (e.g., the set of natural numbers $\mathbb{N}$ is infinite). Since it has a definite truth value (false), it qualifies as a statement.

Conclusion: Yes, it is a statement (a false statement).

Given sentence: "The sun is a star."

Concept: A sentence is a statement if it is either true or false.

Working: It is a scientifically established fact that the Sun is a star. Therefore, this sentence is always true and has a definite truth value.

Conclusion: Yes, it is a statement (a true statement).

Given sentence: "Mathematics is fun."

Concept: A sentence is a statement only if it is either definitely true or definitely false.

Working: This sentence is subjective — for those who enjoy mathematics it may be true, but for others it may not be. It does not have a definite, universal truth value.

Conclusion: No, it is NOT a statement because its truth value varies from person to person.

Given sentence: "There is no rain without clouds."

Concept: A sentence is a statement if it is either true or false.

Working: This is a scientifically true fact — rain requires clouds. The sentence has a definite truth value (true).

Conclusion: Yes, it is a statement (a true statement).

Given sentence: "How far is Chennai from here?"

Concept: Questions, exclamations, and commands are not statements because they cannot be assigned a truth value of true or false.

Working: This is an interrogative sentence (a question). It does not assert anything and cannot be said to be true or false.

Conclusion: No, it is NOT a statement because it is a question and does not have a truth value.

Given statement: "The school is closed if it is a holiday or a Sunday."

Concept:
- Inclusive 'Or': At least one of the two conditions is true (both can be true simultaneously).
- Exclusive 'Or': Exactly one of the two conditions is true (both cannot be true simultaneously).

Working: A day can be both a holiday AND a Sunday at the same time (e.g., a national holiday falling on a Sunday). In either case or both cases, the school remains closed. So both conditions can hold simultaneously.

Conclusion: The "Or" used here is Inclusive Or, because the school is closed when it is a holiday, or a Sunday, or both.

Given statement: "Two lines intersect at a point or are parallel."

Concept: Exclusive 'Or' means exactly one of the two alternatives holds; both cannot be true simultaneously.

Working: Two lines in a plane cannot intersect at a point AND be parallel at the same time. These are mutually exclusive situations — if two lines intersect, they are not parallel, and if they are parallel, they do not intersect.

Conclusion: The "Or" used here is Exclusive Or, because both conditions cannot be true simultaneously.

Given statement: "Students can take French or Sanskrit as their third language."

Concept: Exclusive 'Or' means exactly one of the two alternatives holds.

Working: A student can choose only one language as their third language — either French or Sanskrit, but not both at the same time. The two choices are mutually exclusive.

Conclusion: The "Or" used here is Exclusive Or, because a student cannot take both French and Sanskrit simultaneously as their third language.

Given Statements:
- Some actors are singers. (partial overlap between actors and singers)
- All singers are dancers. (singers ⊂ dancers)

Analysis of Conclusion (1): "Some actors are dancers."
Since some actors are singers, and all singers are dancers, those actors who are singers must also be dancers. Therefore, some actors are dancers. ✓ Conclusion (1) follows.

Analysis of Conclusion (2): "No singer is actor."
Statement I says "Some actors are singers," which directly means some singers ARE actors. So the claim "No singer is actor" is contradicted by Statement I. ✗ Conclusion (2) does not follow.

Answer: (A) — Only conclusion (1) follows.

Given Statements:
- All harmoniums are instruments. (harmoniums ⊂ instruments)
- All instruments are flutes. (instruments ⊂ flutes)

Analysis of Conclusion (1): "All the flutes are instruments."
Statement II says all instruments are flutes, i.e., instruments ⊂ flutes. This does NOT mean all flutes are instruments (the converse is not necessarily true). ✗ Conclusion (1) does not follow.

Analysis of Conclusion (2): "All the harmoniums are flutes."
Since all harmoniums are instruments (Statement I) and all instruments are flutes (Statement II), by transitivity: all harmoniums are flutes. ✓ Conclusion (2) follows.

Answer: (B) — Only conclusion (2) follows.

Given Statements:
- Some mangoes are yellow.
- Some fruits are mangoes.

Analysis of Conclusion (1): "Some mangoes are green."
The statements only tell us that some mangoes are yellow. Nothing is said about mangoes being green. This cannot be concluded from the given statements. ✗ Conclusion (1) does not follow.

Analysis of Conclusion (2): "Fruits are yellow."
Statement II says only SOME fruits are mangoes, and Statement I says only SOME mangoes are yellow. We cannot conclude that all fruits are yellow. ✗ Conclusion (2) does not follow.

Answer: (D) — Neither conclusion (1) nor conclusion (2) follows.