LOGICAL STRUCTURE • Logic - study of reasoning - focuses on the relationship among statements as opposed to the content of any particular statement. Logical methods are used in mathematics to prove theorems and in computer science to prove that programs do what they are alleged to do. ex. All mathematicians wear sandals.
LOGICAL STRUCTURE Propositions – A sentence that is either true or false, but not both. Examples: • 1+1=3 • 2+2=4 - A fact-based declaration is a proposition, even if no one knows whether it is true.
LOGICAL STRUCTURE - A statement cannot be true or false unless it is declarative. This excludes commands and questions. examples: • What time is it? • Go home.
LOGICAL STRUCTURE Definition Let p and q be propositions • The conjunction of p and q, denoted by p q ᴧ , is the proposition p and q • The disjunction of p and q, denoted p ᴠ q , is the proposition p or q
LOGICAL STRUCTURE Compound proposition – combination of propositions Example : p: 1+1 = 3 q: a decade is 10 years then: conjunction: p q ᴧ : 1+1 = 3 and a decade is 10 years. disjunction: p q ᴠ : 1+1 = 3 or a decade is 10
LOGICAL STRUCTURE Each sentence or the propositions has a truth value of either true or false. The truth values of propositions can be described by truth tables.
LOGICAL STRUCTURE Definition The truth value of the compound proposition p q ᴧ is defined by the truth table - states that the conjunction p q ᴧ is true provided that p and q are both true; p q ᴧ is p q p ᴧ q T T T T F F F T F F F F ∧
LOGICAL STRUCTURE Definition The truth value of the compound proposition p q ᴠ is defined by the truth table - states that the disjunction p q ᴠ is true if either p or q or both are true; false if both p and q are false p q p q ᴠ T T T T F T F T T F F F
LOGICAL STRUCTURE Definition The negation of p denoted by ¬p is the proposition not p. The truth value of the proposition ¬p is defined by the truth table In other words, the negation of a proposition has the opposite truth value from the proposition itself. p ¬p T F F T
LOGICAL STRUCTURE Example p : Blaise Pascal invented several calculating machines. q : The first all-electronic digital computer was constructed in the 20th century. r : П was calculated to 1M decimal digits in 1954
LOGICAL STRUCTURE • represent the proposition: Either Blaise Pascal invented several calculating machines and it is not the case that the first all electronic digital computer was constructed in the 20th century; or П was calculated to 1M decimal digits in 1954.
LOGICAL STRUCTURE Definition: If p and q are propositions, the compound propositions if p then q is called conditional proposition and is denoted by p→ q. • The proposition p is called the hypothesis (or antecedent) and the proposition q is called the conclusion (or consequent)
LOGICAL STRUCTURE example: p: The Math Dept. gets an additional Php 200,00.
- Winter '99
- Logic, logical structure