Discrete Mathematics : Proofs, Structures and Applications, by Rowan Garnier

By Rowan Garnier

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 equipment of facts Mathematical Induction units units and MembershipSubsetsOperations Read more...

summary: 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 good judgment Mathematical facts the character of facts Axioms and Axiom platforms equipment of evidence Mathematical Induction units units and MembershipSubsetsOperations on SetsCounting TechniquesThe Algebra of units households of units The Cartesian Product varieties and Typed Set TheoryRelations family and Their Representations homes of kin

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11 p¯ ∧ q p∨q p¯ → q q ↔ p. Consider the propositions: p : Mary laughs. q : Sally cries. r : Jo shouts. Write in words the following compound propositions: (i) (ii) (iii) (iv) (v) p → (q r) (r ∧ q) ↔ p (p → q¯) ∧ (r → q) p ∨ (¯ q ∨ r¯) (p ∨ r) ↔ q¯. Logic 12 3. Let p, q and r denote the following propositions: p : Bats are blind. q : Gnats eat grass. r : Ants have long teeth. Express the following compound propositions symbolically. (i) (ii) (iii) (iv) If bats are blind then gnats don’t eat grass.

Premises : Conclusion : p → q, r → s, q¯, r p¯ ∧ s As with the previous example, we shall often find it useful to work backwards by asking ourselves what needs to be added to the list to justify adding the conclusion. Here the conclusion is a conjunction. If the list were to contain each of the conjuncts p¯ and s, then we have a rule of inference (Conjunction) which will allow us to add p¯∧ s. In this example, each of the conjuncts follows directly from a rule of inference applied to a pair of premises.

Pn is true. Now if (P1 ∧ P2 ∧ . . ∧ Pn ) Q, then Q is true whenever P1 ∧ P2 ∧ . . e. when P1 , P2 , . . and Pn are all true. This means that we can add to our list any proposition which is logically implied by the conjunction of a set of earlier propositions in the sequence. The relationship (P1 ∧ P2 ∧ . . ∧ Pn ) Q means that we can regard Q as the conclusion of a valid argument with premises P1 , P2 , . . Pn . ) Hence a justification for adding a proposition Logic 30 to the sequence is that it is the conclusion of a valid argument whose premises are already included in the list.

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