3.2 Truth Tables for Negation, Conjunction, and Disjunction 109 example illustrates, an and statement is true only when both simple statements are true. The results are summarized in Table 3.4, the truth table for the conjunction. Civil Technician Municipal program for redevelopment seeks on-site technician. The applicant must have a two-year college degree in civil technology or five years of related experience. Interested candidates please call 555-1234. The conjunction ∧ p q is true only when both p and q are true. Disjunction Consider the job description in the margin that describes two job requirements. Who qualifies for the job? To help analyze the statement, translate it into symbolic form. Let p be “A requirement for the job is a two-year college degree in civil technology” and q be “A requirement for the job is five years of related experience.” The statement in symbolic form is ∨ p q. For the two simple statements, there are four distinct cases (see Table 3.5). CASE 1: p is true and q is true. A candidate has a two-year college degree in civil technology and five years of related experience. The candidate has both requirements and qualifies for the job. Consider qualifying for the job as a true statement and not qualifying as a false statement. Since the candidate qualifies for the job, we put a T in the ∨ p q column. CASE 2: p is true and q is false. A candidate has a two-year college degree in civil technology but does not have five years of related experience. The candidate still qualifies for the job with only the two-year college degree. Thus, we put a T in the ∨ p q column. CASE 3: p is false and q is true. The candidate does not have a two-year college degree in civil technology but does have five years of related experience. The candidate still qualifies for the job with only the five years of related experience. Thus, we put a T in the ∨ p q column. CASE 4: p is false and q is false. The candidate does not have a two-year college degree in civil technology and does not have five years of related experience. The candidate does not meet either of the two requirements and therefore does not qualify for the job. Thus, we put an F in the ∨ p q column. In examining the four cases, we see that there is only one case in which the candidate does not qualify for the job: case 4 (FF). As this example indicates, an or statement will be true in every case, except when both simple statements are false. The results are summarized in Table 3.5, the truth table for the disjunction. Table 3.5 Disjunction p q p q ∨ T T T T F T F T T F F F The disjunction ∨ p q is true when either p is true, q is true, or both p and q are true. The disjunction ∨ p q is false only when p and q are both false.
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