Lesson 4.1 Introduction to Logic

• Logic; the study of method and principles used in distinguishing correct from incorrect arguments
• Proposition or Statement; a declarative sentence which is either true or false, but not both
• Truth Value; the truth or falsity of a statement
• Propositional variables are used to represent propositions, usually denoted by p, q, r , s and t
• Open Sentence; contains one or more variables, that is, it is either true or false depending on the value of the placeholder
• Closed Sentence, a mathematical sentence that is known to be either true or false
• Compound Proposition; a proposition formed from simple propositions using logical connectors or some combinations of logical connectors
• Simple Proposition; if cannot be broken down any further into other component proposition
• Quantifiers; words, expression, or phrases that point out the number of elements that a statement relates to
• Two types of quantifiers;

  1. Universal Quantifier
  2. Existential Quantifier
• R; set of real numbers
• N; set of natural numbers
• Z; set of integers
• Negations; If p is true then not p is false. If p is false then not p is true
• Conjunctions; a compound statement connecting two proposition with the word "and"
• Conjunctions; if p and q are true, then p ^q is true, else false
• Disjunctions; a compound statement formed by connecting two statements with the word "or"
• Disjunction; if p and q are false, then pvq is false otherwise true
• A compound statement formed by connecting two statements with the words "if...,then".
• Biconditional; A compound statement formed by connecting two statements with the words "if and only if
• Exclusive-or; compound proposition "p exclusive-or q"
• exclusive-or'; if p and q are true both false, then p and q is false; if p and q have opposite truth values, then p and q are true
• Truth Table; a mathematical table used to determine if a compound statement is true or false
• Basic Logic Operators;

  1. Negation
  2. Conjunction
  3. Disjunction
  4. Conditional
  5. Biconditional
  6. Exclusive-or
• Tautology; a compound statement that is always true, regardless oof the truth values of its components
• Contradiction; a compound proposition that is always false
• Contingency; neither a tautology nor a contradiction
• Equivalent Statements; statements whose truth values is always either true or both false whenever they have identical truth tables
• Conditional Propositions;

  1. Converse
  2. Contrapositive
  3. Inverse
• Argument; an assertion that a given serios called premises yields (has consequence) another statement Q, called the conclusion
• Premises; are intended to demonstrate or at least provide some evidences for the conclusion
• Valid; when all the premises are true if forces the conclusion to be true
• Invalid Argument or Fallacy; an argument which is not valid
• Law of Detachment "modus ponens"; if p then q, an argument is valid if the statement is a tautology
• Valid;

  1. Law of Detachment "Modus Ponens"
  2. Law of Contraposition "Modus Tollens"
  3. Law of Syllogism
  4. Rule of Disjunctive Syllogism