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 facts of the Validity of Arguments Predicate good judgment Arguments in Predicate common sense Mathematical evidence the character of facts Axioms and Axiom platforms tools of facts Mathematical Induction units units and MembershipSubsetsOperations on SetsCounting TechniquesThe Algebra of units households of units The Cartesian Product varieties and Typed Set TheoryRelations relatives and Their Representations homes of Relations. 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 facts of the Validity of Arguments Predicate good judgment Arguments in Predicate good judgment Mathematical facts the character of facts Axioms and Axiom platforms tools of facts Mathematical Induction units units and MembershipSubsetsOperations on SetsCounting TechniquesThe Algebra of units households of units The Cartesian Product varieties and Typed Set TheoryRelations family members and Their Representations homes of kin
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Additional resources for Discrete Mathematics : Proofs, Structures and Applications, Third Edition
4. p q (2. Simp) (1, 3. MP) Note that the propositions in the sequence are numbered so that they can be referred to later and also that a justification must be provided for the addition of each proposition to the list. Putting the steps together, the complete formal proof is the following. 1. 2. 3. 4. 2. p→q p∧r p q (premise) (premise) (2. Simp) (1, 3. MP) Provide a formal proof of the validity of the following argument: Premises : Conclusion : p → q¯, q ∨ r p→r Note that, if we could add q¯ → r to our sequence, we could apply the Hypothetical Syllogism rule to this and the first premise and thereby justify the addition of the conclusion.
Two propositions as similar as ‘Bill has green eyes’ and ‘Jeff has green eyes’ would have to be symbolized by p and q. We have as yet no means of expressing the fact that both propositions are about ‘green eyes’. A predicate describes a property of one or several objects or individuals. Examples of predicates might be: (a) (b) (c) (d) (e) . . is red. . has long teeth. . enjoys standing on his head. . has spiky leaves. . cannot be tolerated under any circumstances. The space in front of these predicates can be filled in with the names of objects or individuals where appropriate to form a proposition which may be true or false in the usual way.
We return to this point after we have considered the two forms. Given the two propositions p and q, p ∨ q symbolizes the inclusive disjunction of p and q. This compound proposition is true when either or both of its components are true and is false otherwise. Thus the truth table for p ∨ q is given by: p q p∨q T T F T F T T T T F F F 6 Logic The exclusive disjunction of p and q is symbolized by p q. e. one or other, but not both) of its components is true. The truth table for p q is given by: p q p q T T F T F T F T T F F F When two simple propositions are combined using ‘or’, context will often provide the clue as to whether the inclusive or exclusive sense is intended.
Discrete Mathematics : Proofs, Structures and Applications, Third Edition by Rowan Garnier