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 good judgment 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 varieties and Typed Set TheoryRelations relatives and Their Representations houses of Relations. 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 common sense Mathematical evidence the character of evidence Axioms and Axiom structures equipment of facts Mathematical Induction units units and MembershipSubsetsOperations on SetsCounting TechniquesThe Algebra of units households of units The Cartesian Product forms and Typed Set TheoryRelations family and Their Representations houses of family members
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Additional info for Discrete Mathematics : Proofs, Structures and Applications, Third Edition
The substitution rule and the replacement laws give us a means of establishing logical equivalences between propositions without drawing up a truth table. We demonstrate this in the following example. 9 Prove that (¯ p ∧ q) ∨ (p ∨ q) ≡ p¯. Solution (¯ p ∧ q) ∨ (p ∨ q) ≡ (¯ p ∧ q) ∨ (¯ p ∧ q¯) ≡ p¯ ∧ (q ∨ q¯) ≡ p¯ ∧ t ≡ p¯. 4 1. 9. (i) (ii) (iii) (iv) (v) 2. (p ∧ p) ∨ (¯ p ∨ p¯) ≡ t. (p ∧ q) ∧ q ≡ p ∧ q. p → q ≡ p ∧ q¯. (p ∧ q) → r ≡ (¯ p ∨ q¯) ∨ r. q ∧ [(p ∨ q) ∧ (¯ q ∧ p¯)] ≡ q ∧ (q ∨ p). 9 to show that p ∧ (q ∨ p¯) is logically equivalent to p ∧ q.
Converse: q → p: If Sara will sing then Jack plays his guitar. Inverse: p¯ → q¯: If Jack doesn’t play his guitar then Sara won’t sing. Contrapositive: q¯ → p¯: If Sara won’t sing then Jack doesn’t play his guitar. As we have shown, ‘If Jack plays his guitar then Sara will sing’ and ‘If Sara won’t sing then Jack doesn’t play his guitar’ are equivalent propositions. 3 1. Prove that (p → q) ≡ (¯ p ∨ q). 2. Prove that (p ∧ q) and (p → q¯) are logically equivalent propositions. 3. Prove that (p 4. Prove that p logically implies (q → p).
10. Consider a new connective, denoted by |, where p|q is defined by the following truth table: p q p|q T T F T F T F T T F F T Show that: (i) (ii) p¯ ≡ (p|p) (p ∧ q) ≡ (p|q)|(p|q). 9 above to deduce that a proposition involving any of the five familiar connectives can be written in a logically equivalent form which uses only the connective denoted by |. 11. State the converse, inverse and contrapositive of the proposition: ‘If it’s not Sunday then the supermarket is open until midnight’. 4. ) These are often referred to as ‘replacement laws’ because, as we shall see later, there are situations where it is useful to substitute one proposition for another logically equivalent form.