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AND. Chapter 3. Logic. WHAT YOU WILL LEARN. • Statements, quantifiers, and compound statements • Statements involving the words not , and , or , if… then… , and if and only if

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  1. AND

  2. Chapter 3 Logic

  3. WHAT YOU WILL LEARN • Statements, quantifiers, and compound statements • Statements involving the words not, and, or, if… then…, and if and only if • Truth tables for negations, conjunctions, disjunctions, conditional statements, and biconditional statements • Self contradictions, tautologies, and implications

  4. WHAT YOU WILL LEARN • Equivalent statements, De Morgan’s Law, and variations of conditional statements • Symbolic arguments and standard forms of arguments • Euler diagrams and syllogistic arguments • Using logic to analyze switching circuits

  5. Section 1 Statements and Logical Connectives

  6. HISTORY—The Greeks: • Aristotelian logic: The ancient Greeks were the first people to look at the way humans think and draw conclusions. Aristotle (384-322 B.C.) is called the father of logic. This logic has been taught and studied for more than 2000 years.

  7. Mathematicians • Gottfried Wilhelm Leibniz (1646-1716) believed that all mathematical and scientific concepts could be derived from logic. He was the first to seriously study symbolic logic. In this type of logic, written statements use symbols and letters. • George Boole (1815 – 1864) is said to be the founder of symbolic logic because he had such impressive work in this area.

  8. Logic and the English Language • Connectives - words such as and, or, if, then • Exclusive or - one or the other of the given events can happen, but not both. • Inclusive or - one or the other or both of the given events can happen.

  9. Statements and Logical Connectives • Statement - A sentence that can be judged either true or false. • Labeling a statement true or false is called assigning a truth value to the statement. • Simple Statements - A sentence that conveys only one idea and can be assigned a truth value. • Compound Statements - Sentences that combine two or more simple statements.

  10. Negation of a Statement • Negation of a statement – change a statement to its opposite meaning. • The negation of a false statement is always a true statement. • The negation of a true statement is always a false statement.

  11. Quantifiers • Quantifiers - words such as all, none, no, some, etc… • Be careful when negating statements that contain quantifiers.

  12. None are. Some are not. All are. Some are. Negation of Quantified Statements Form of negation Some are not. Some are. None are. All are. Form of statement All are. None are. Some are. Some are not.

  13. Example: Write Negations Write the negation of the statement. Some candy bars contain nuts. Solution: Since some means “at least one” this statement is true. The negation is “No candy bars contain nuts,” which is a false statement.

  14. Example: Write Negations continued Write the negation of the statement. All tables are oval. Solution: This is a false statement since some tables are round, rectangular, or other shapes. The negation would be “Some tables are not oval.”

  15. Compound Statements • Statements consisting of two or more simple statements are called compound statements. • The connectives often used to join two simple statements are and, or, if…then…, andif and only if.

  16. Not Statements (Negation) • The symbol used in logic to show the negation of a statement is ~. It is read “not”. • Example: The negation of p is: ~ p.

  17. And Statements (Conjunction) • is the symbol for a conjunction and is read “and.” • The other words that may be used to express a conjunction are: but, however, and nevertheless. • Example: The conjunction of p and q is: p ^ q.

  18. Example: Write a Conjunction Write the following conjunction in symbolic form: The dog is gray, but the dog is not old. Solution: Let p and q representthe simple statements. p: The dog is gray. q: The dog is old. In symbolic form, the compound statement is pΛ ~ q

  19. Or Statements (Disjunction) • The disjunction is symbolized by and read “or.” • In this book the “or” will be the inclusive or (except where indicated in the exercise set). • Example: The disjunction of p and q is: p q.

  20. Example: Write a Disjunction Write the statement in symbolic form. Carl will not go to the movies or Carl will not go to the baseball game. Solution: Let p and q representthe simple statements. p: Carl will go to the movies. q: Carl will go to the baseball game. In symbolic form, the compound statement is

  21. If-Then Statements (continued) • The conditional is symbolized by and is read “if-then.” • The antecedent is the part of the statement that comes before the arrow. • The consequent is the part that follows the arrow. • Example: If p, then q is symbolized as: p  q.

  22. Solutionsa) b) Example: Write a Conditional Statement Let p: Nathan goes to the park. q: Nathan will swing. Write the following statements symbolically. a. If Nathan goes to the park, then he will swing. b. If Nathan does not go to the park, then he will not swing.

  23. If and Only If Statements (Biconditional) • The biconditional is symbolized by and is read “if and only if.” • If and only if is sometimes abbreviated as “iff.”

  24. CourseSmart Page 109

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