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By Rowan Garnier

Common sense Propositions and fact Values Logical Connectives and fact Tables Tautologies and Contradictions Logical Equivalence and Logical Implication The Algebra of Propositions Arguments Formal evidence of the Validity of Arguments Predicate common sense Arguments in Predicate common sense Mathematical evidence the character of evidence Axioms and Axiom platforms tools of evidence Mathematical Induction units units and MembershipSubsetsOperations Read more...

summary: good judgment Propositions and fact Values Logical Connectives and fact Tables Tautologies and Contradictions Logical Equivalence and Logical Implication The Algebra of Propositions Arguments Formal facts of the Validity of Arguments Predicate common sense Arguments in Predicate good judgment Mathematical facts the character of facts Axioms and Axiom structures tools of evidence Mathematical Induction units units and MembershipSubsetsOperations on SetsCounting TechniquesThe Algebra of units households of units The Cartesian Product forms and Typed Set TheoryRelations kin and Their Representations houses of kinfolk

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Extra info for Discrete Mathematics : Proofs, Structures and Applications, Third Edition

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1. 2. 3. 4. 5. 6. 7. 4. p→q r→s q¯ r p¯ s p¯ ∧ s (premise) (premise) (premise) (premise) (1, 3. MT) (2, 4. MP) (5, 6. Conj) Give a formal proof for the following argument: Premises : Conclusion : p ∧ (q ∨ r), p¯ ∨ q¯ r∨q The proof here is a little longer and, again, it helps to work backwards. The conclusion here is the inclusive disjunction of r and q. Note that the Addition rule would allow us to add the proposition r ∨ q if our list contained simply the proposition r. So let’s concentrate on how we might justify the inclusion of r.

The summer is hot and windy. Therefore my garden flourishes. (v) People are happy if and only if they are charitable. Nobody is both happy and charitable. Hence people are unhappy and uncharitable. Predicate Logic (vi) 35 If you go to college or get a good job then you will be successful and respected. You go to college. Therefore you will be respected. (vii) If I eat cheese I get sick and if I drink wine I get sick. If I go to Ira’s I eat cheese or I drink wine. I’m going to Ira’s. Therefore I shall get sick.

Otherwise the argument is said to be invalid. Thus if we have premises P1 , P2 , . . e. if (P1 ∧ P2 ∧ · · · ∧ Pn ) → Q is a tautology. 4) is that whenever P1 , P2 , . . , Pn are all true, then Q must be true. This makes sense since it ensures that, in a valid argument, a set of premises all of which are true cannot lead to a false conclusion. 10 1. Test the validity of the following argument: ‘If you insulted Bob then I’ll never speak to you again. ’ Solution We define: p: You insulted Bob. q: I’ll never speak to you again.

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