Example 142 Investigate the validity of the following argument using a Flow of

Example 142 investigate the validity of the following

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Example 1.4.2. Investigate the validity of the following argument using (a) Flow of the argument, (b) by using truth tables. 13
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(a) P Q R P ¬ R Q (a) Flow of the argument, If ¬ R is true then R is false. If R is false and R P is true then P is true. If P is true then for P Q to be true then Q must be true. Then we deduce that the argument is valid . (b) by using truth tables, we need to test whether the statement: [( P Q ) ( R P ) ( ¬ R )] Q is tautology. Let A ( P Q ) ( R P ) ( ¬ R ) in order to simplify our table. P Q R ¬ R P Q R P A ( P Q ) ( R P ) ( ¬ R ) A Q 1 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1 Example 1.4.3. Investigate the validity of the following argument using (a) Flow of the argument, 14
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(b) by using truth tables. (a) P ¬ Q R Q ¬ P R (a) Flow of the argument, If ¬ P is true then P is false. If P is false and P ∨ ¬ Q is true then ¬ Q is true. If ¬ Q is true, then Q is false. If Q is false and R Q is true then R is false. Since the conclusion is given R which in this instant we found it to be false, then we deduce that the argument is invalid or it is a fallacy. (b) by using truth tables, we need to test whether the statement: [( P ∨ ¬ Q ) ( R Q ) ( ¬ P )] R is tautology. Let A ( P ∨ ¬ Q ) ( R Q ) ( ¬ P ) in order to simplify our table. P Q R ¬ P ¬ Q P ∨ ¬ Q R Q A ( P ∨ ¬ Q ) ( R Q ) ( ¬ P ) A R 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 0 15
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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 Q P R ¬ R Q (a) Flow of the argument, If ¬ R is true then R is false. If P R is true and R is false then P is false. If P Q is true and P is false then Q is true. Therefore the argument is valid. (b) We leave it as an exercise to students to verify using truth table that the argument 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
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(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 Q Q R ¬ R ¬ P (a) Flow of the argument, If ¬ R is true then R is false. If Q R is true and R is false then Q is true. If P Q is true and Q is true then P is either true or false hence 6 P is also either true or false . So we cannot affirm that 6 P is true. Therefore the argument is invalid .
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