Scholarship Rules and the Truth-Table Check
At a higher secondary school, the scholarship committee has 40 applicants and publishes three rules: (1) If an applicant submits the income certificate, then the applicant attends the interview. (2) An applicant attends the interview only if the applicant scores above 60 in the written test. (3) An applicant receives the scholarship only if the applicant both submits the income certificate and scores above 60. Let C = submits the income certificate, I = attends the interview, S = scores above 60, and W = receives the scholarship. A teacher tells the assembly: 'From rules 1 and 2 it follows that every certificate submitter scores above 60.' A second teacher claims: 'Rule 3 is just the converse: anyone with the certificate and above 60 must receive the scholarship.' A student says: 'Rules 1 and 2 mean either you submit the certificate or you score above 60, but not both.' The committee has one afternoon to settle the logic before appeals.
1. Symbolize rules 1-3 and the first teacher's conclusion. Use a truth table to decide whether the conclusion follows.
Read the worked answer
Symbolization: Rule 1 is C → I. Rule 2 is I → S, because 'I only if S' means that S is necessary for I. Rule 3 is W → (C ∧ S), because receiving the scholarship requires both conditions. The first teacher's conclusion is C → S.
The conclusion follows by hypothetical syllogism: from C → I and I → S we get C → S. A truth table confirms there is no counterexample. In the eight rows for C, I, S, the premises (C → I) and (I → S) are jointly true only when C, I, and S are all true or when C is false and I and S are both true; in those rows C → S is also true. The only rows that would make C → S false are C true and S false, but then either I is false, making C → I false, or I is true, making I → S false. So the premises cannot be true while the conclusion is false.
2. Test the second teacher's claim that rule 3 is equivalent to (C ∧ S) → W.
Read the worked answer
The claim is false. Rule 3 says W → (C ∧ S); the second teacher asserts the converse (C ∧ S) → W. A conditional and its converse are not equivalent.
A single row refutes equivalence: let C = true, S = true, W = false. Then W → (C ∧ S) is true, because a false antecedent makes the conditional true. But (C ∧ S) → W is false, because the antecedent C ∧ S is true and the consequent W is false. In ordinary terms, a student may submit the certificate and score above 60 yet still not receive the scholarship-for example, because of a quota, a disciplinary bar, or a missing document. Rule 3 states a necessary condition, not a sufficient condition.
3. Evaluate the student's claim that rules 1 and 2 mean 'either C or S, but not both'.
Read the worked answer
The student has confused a conditional with an exclusive disjunction. Rules 1 and 2 together yield C → S, which is equivalent to ¬C ∨ S. The student's formula is (C ∨ S) ∧ ¬(C ∧ S), an exclusive or.
They differ on the row C = true, S = true: C → S is true, but the exclusive or is false because both disjuncts are true. They also differ in meaning: C → S permits both certificate submission and a score above 60, while the exclusive or forbids that combination. The student's reading would wrongly make it impossible for a strong applicant both to submit the certificate and to score above 60. So the claim is not a valid paraphrase of rules 1 and 2.