MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Learning Goals 1. Understand the meaning of necessary and sufficient conditions (carried over from Wednesday). 2. Understand the difference between a valid argument and an invalid argument. 3. Be able to use a truth table to determine the validity of an argument. 4. Be able to use rules of inference to determine the validity of an argument. 5. Apply the process of looking for truth values that would demonstrate that an argument is invalid. Issues raised in Question 5 of the quiz: Using the rules of inference. rying to find truth values that make all premises true but conclusion false. he textbook says a sound argument is a valid argument with true premises. Can you have a valid argument with premises that are not true? Can you prove Q3 from the quiz using rules of inference? Can you explain Q4 from the quiz?
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Necessary and sufficient conditions (carried over from Wednesday): Activity 1: Let nn be a positive integer. (a) State a condition that is necessary, but not sufficient, for nn to be divisible by 6. (b) State a condition that is sufficient, but not necessary, for nn to be divisible by 6. (c) State a necessary and sufficient condition for nn to be divisible by 6.
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Argument form: Activity 2: Consider the following argument: If wages are raised, buying increases. If there is a depression, buying does not increase. herefore there is not a depression or wages are not raised. Let ww represent the statement wages are raised, bb represent the statement buying increases, and dd represent the statement there is a depression. Write this argument in symbolic form. w bb dd bb ( dd ww)
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Proving an argument is valid (truth table method): ww bb dd bb ww dd bb ww dd bb dd ww Question 1 (Assessed): his argument is: A. valid. B. Invalid.
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Proving an argument is valid (rules of inference method): Activity 4: Use the laws of logical equivalence and the rules of inference to show that this argument is valid. 1: ww bb 2: dd bb 3: bb dd from 2 by taking the contrapositive 4: ww dd from 1 and 3 by the ransitivity argument form 5: ww dd from 4 by rewriting from 5 by commutative law
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Q3 from the L5 pre-class quiz Premise 1: pp qq Premise 2: pp rr Premise 3: pp rr Conclusion: qq 1: pp qq 2: pp rr 3: pp rr 4: pp rr (pp rr) from 2 and 3 5: pp (rr rr) from 4 by the Distributive law 6: pp cc from 5 by the Negation law 7: pp from 6 by the Identity law Conclusion: qq from 1 and 7 by Modus Ponens
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Proving an argument is valid (by checking if the argument is invalid): Activity 5: ry to find truth values for the statement variables that make all the premises true but the conclusion false (and hence demonstrate that the argument is invalid). or this to be false we require dd to be false and ww to be false. hus we have dd true and ww true. or this to be true, given that we already know that ww is true, we require bb to be true. hus we have bb true. or this to be true, given that we already know that dd is true, we require bb to be true. hus we have bb false. his is a contradiction, so it is not possible to find truth values for the variables ww, bb, dd that make all the premises true and the conclusion false, so the argument is valid.
MAH1061/MAH7861 Discrete Mathematics Semester 2, 2016 Q4 from the L5 pre-class quiz Premise 1: pp qq Premise 2: qq rr Premise 3: pp qq Conclusion: rr ind truth values for the statement variables that make all the premises true but the conclusion false (and hence demonstrate that the argument is invalid). Conclusion: rr Premise 2: qq rr Premise 1: pp qq or this to be false we require rr false. or this to be true, given that we already know that rr is false, we require qq false. or this to be true, given that we already know that qq is false, we require pp false. Premise 3: pp qq We already know that pp is false and qq is false. Hence this premise is true. So with pp false, qq false and rr false, the premises are all true and the conclusion is false, so these truth values demonstrate that the argument is invalid.