discrete-structures - DISCRETE STRUCTURES LOGICAL STRUCTURE Logic study of reasoning focuses on the relationship among statements as opposed to the

discrete-structures - DISCRETE STRUCTURES LOGICAL STRUCTURE...

This preview shows page 1 - 15 out of 100 pages.

DISCRETE STRUCTURES
Image of page 1
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.
Image of page 2
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.
Image of page 3
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.
Image of page 4
LOGICAL STRUCTURE Definition Let p and q be propositions The conjunction of p and q, denoted by p q , is the proposition p and q The disjunction of p and q, denoted p q , is the proposition p or q
Image of page 5
LOGICAL STRUCTURE Compound proposition – combination of propositions Example : p: 1+1 = 3 q: a decade is 10 years then: conjunction: p q : 1+1 = 3 and a decade is 10 years. disjunction: p q : 1+1 = 3 or a decade is 10
Image of page 6
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.
Image of page 7
LOGICAL STRUCTURE Definition The truth value of the compound proposition p q is defined by the truth table - states that the conjunction p q is true provided that p and q are both true; p q is p q p q T T T T F F F T F F F F
Image of page 8
LOGICAL STRUCTURE Definition The truth value of the compound proposition p q is defined by the truth table - states that the disjunction p q is true if either p or q or both are true; false if both p and q are false p q p q T T T T F T F T T F F F
Image of page 9
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
Image of page 10
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
Image of page 11
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.
Image of page 12
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)
Image of page 13
LOGICAL STRUCTURE example: p: The Math Dept. gets an additional Php 200,00.
Image of page 14
Image of page 15

You've reached the end of your free preview.

Want to read all 100 pages?

  • Winter '99
  • Averróis
  • Logic, logical structure

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes