Example 1.4.2.Investigate the validity of the following argument using(a) Flow of the argument,(b) by using truth tables.13
(a)P⇒QR∨P¬R∴Q(a) Flow of the argument,If¬Ris true thenRis false. IfRis false andR∨Pis true thenPis true.IfPis true then forP⇒Qto be true thenQmust be true. Then wededuce that the argument isvalid.(b) by using truth tables,we need to test whether the statement:[(P⇒Q)∧(R∨P)∧(¬R)]⇒Qis tautology.LetA≡(P⇒Q)∧(R∨P)∧(¬R) in order to simplify our table.PQR¬RP⇒QR∨PA≡(P⇒Q)∧(R∨P)∧(¬R)A⇒Q1110110111011111101001011001010101101101010110010010110100011001Example 1.4.3.Investigate the validity of the following argument using(a) Flow of the argument,14
(b) by using truth tables.(a)P∨¬QR⇒Q¬P∴R(a) Flow of the argument,If¬Pis true thenPis false. IfPis false andP∨ ¬Qis true then¬Qistrue. If¬Qis true, thenQis false. IfQis false andR⇒Qis true thenRis false. Since the conclusion is givenRwhich in this instant we found itto be false, then we deduce that the argument isinvalidor it is a fallacy.(b) by using truth tables,we need to test whether the statement:[(P∨ ¬Q)∧(R⇒Q)∧(¬P)]⇒Ris tautology.LetA≡(P∨ ¬Q)∧(R⇒Q)∧(¬P) in order to simplify our table.PQR¬P¬QP∨ ¬QR⇒QA≡(P∨ ¬Q)∧(R⇒Q)∧(¬P)A⇒R11100110111000110110101100110001110101110010101010010100111100100011111015
Looking at the last column, we can clearly deduce the statement is not tautology,hence the argument is not valid just as we expected.Example 1.4.4.Investigate the validity of the following argument using(a) Flow of the argument,(b) by using truth tables.(a) Mary plays netball or volleyball at school.If Mary plays netball at school then she goes to Church.Mary doesn’t go to Church.∴Mary plays volleyball.Now let,P: Mary plays netball at school.Q: Mary plays volleyball at school.R: Mary goes to church.P∨QP⇒R¬R∴Q(a) Flow of the argument,If¬Ris true thenRis false. IfP⇒Ris true andRis false thenPis false.IfP∨Qis true andPis false thenQis true. Therefore the argument isvalid.(b) We leave it as an exercise to students to verify using truth table that theargument is valid by verifying the statement is tautology.Example 1.4.5.Investigate the validity of the following argument using(a) Flow of the argument,16
(b) by using truth tables.(a) If Mpho eats mangos then he lives in Venda.Mpho either lives in Venda or Soweto.Mpho does not live in Soweto.∴Mpho doesn’t eat mangos.Now let,P: Mpho eats mangos.Q: Mpho lives in Venda.R: Mpho lives in Soweto.P⇒QQ∨R¬R∴¬P(a) Flow of the argument,If¬Ris true thenRis false. IfQ∨Ris true andRis false thenQis true.IfP⇒Qis true andQis true thenPiseither true or falsehence6Pisalsoeither true or false. So we cannot affirm that6Pis true. Thereforetheargument is invalid.