Week 2 Logical Equivalence and Logical Implication

# Week 2 Logical Equivalence and Logical Implication -...

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Unformatted text preview: CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE & SOFTWARE ENGINEERING COMP 232/2 Mathematics for Computer Science FALL 2010 Logical Equivalence and Logical Implication Everything is vague to a degree you do not realize till you have tried to make it precise. — Bertrand Russell 1. Statements 1.1 We use the word “statement” to mean declarative statement. Such a state- ment is true , e.g. , Quebec is adjacent to Ontario; or false , e.g. , Quebec is adjacent to Saskatchewan; or may depend on one or more variables, e.g. , Province X is adjacent to province Y . 1.2 Statements of the first two types are called propositions ; statements of the third type are called propositional functions (or predicates ). 2. Definitions: def ≡ 2.1 We write P def ≡ Q to mean Let P be defined to be ( the statement ) Q . 2.2 For example, the definition P ( x ) def ≡ x > 3 means Let P ( x ) be the statement “ x > 3 ”. 2.3 Logical Operators: ¬ , ∧ , ∨ , → , ↔ , ↑ , ↓ , , ⊕ negation ( “not” ) ¬ P def ≡ It is not the case that P . conjunction ( “and” ) P ∧ Q def ≡ Both P and Q . disjunction ( “or” ) P ∨ Q def ≡ Either P or else Q . conditional ( “imp” ) P → Q def ≡ ( P ∧ Q ) ∨ ( ¬ P ∧ Q ) ∨ ( ¬ P ∧ ¬ Q ) . biconditional ( “bimp” ) P ↔ Q def ≡ ( P ∧ Q ) ∨ ( ¬ P ∧ ¬ Q ) . anticonjunction ( “nand” ) P ↑ Q def ≡ ( P ∧ ¬ Q ) ∨ ( ¬ P ∧ Q ) ∨ ( ¬ P ∧ ¬ Q ) . antidisjunction ( “nor” ) P ↓ Q def ≡ ( ¬ P ∧ ¬ Q ) . anticonditional ( “nimp” ) P Q def ≡ ( P ∧ ¬ Q ) . antibiconditional ( “xor” ) P ⊕ Q def ≡ ( P ∧ ¬ Q ) ∨ ( ¬ P ∧ Q ) . 2.4 Quantifiers: ∀ , ∃ universal ∀ xP ( x ) def ≡ P ( x ) is true for every value of x . existential ∃ xP ( x ) def ≡ P ( x ) is true for at least one value of x . 2.5 Example. The logical operators could also be defined with truth tables. P Q P P ∧ Q P ∨ Q P → Q P ↔ Q T T T T T T T T F T F T F F F T F F T T F F F F F F T T P Q ¬ P P ↑ Q P ↓ Q P Q P ⊕ Q T T F F F F F T F F T F T T F T T T F F T F F T T T F F Logical Equivalence & Logical Implication 1 19-Sep-2010, 2:02 pm 2.6 Conditional Statements. The conditional statement P → Q has the same truth table as ¬ P ∨ Q and thus means Either P is false or else Q is true. For example, the statement Either you cook that pigeon or else I won’t eat it could be written You do not cook that pigeon → I won’t eat it . 3. Logical Equivalence: ⇐⇒ 3.1 Definition. We write P ⇐⇒ Q to mean that the statements P and Q always have the same truth value, regardless of the values of any variables that may appear in them. 3.2 For example (assuming x is restricted to the set of real numbers): x > 3 ⇐⇒ 3 < x ; x 2 > 5 ⇐⇒ x > √ 5 or x <- √ 5 ....
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