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Chapter 1

Chapter 1. The Logic of Compound Statements. Section 1.1. Logical Form and Logical Equivalence. Statements. A statement is a sentence that is either true or false, but not both. Statements: It is raining . I am carrying an umbrella. Not statements He has a driver’s license.

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Chapter 1

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  1. Chapter 1 The Logic of Compound Statements

  2. Section 1.1 Logical Form and Logical Equivalence

  3. Statements • A statement is a sentence that is either true or false, but not both. • Statements: • It is raining. • I am carrying an umbrella. • Not statements • He has a driver’s license. • Are you there? • x + y > 0

  4. Logical Operators • Binary operators • Conjunction – “and”. • Disjunction – “or”. • Unary operator • Negation – “not”. • Other operators • XOR – “one or the other but not both”” • NAND – “not both” • NOR – “neither”

  5. Logical Symbols • Statements are represented by letters: p, q, r, etc. •  means “and”. •  means “or”. •  means “not”.

  6. Examples • Basic statements • p = “It is raining.” • q = “I am carrying an umbrella.” • Compound statements • pq = “It is raining and I am carrying an umbrella.” • pq = “It is raining or I am carrying an umbrella.” • p = “It is not raining.”

  7. Examples • But compound statements • “it is not hot but it is sunny.” • but in this case is ^ “and” • p = “it is not hot” • q = “it is sunny” • expression: p ^ q • “it is not hot and it is not sunny”

  8. Truth Table of an Expression • Make a column for every variable. • List every possible combination of truth values of the variables. • Make one more column for the expression. • Write the truth value of the expression for each combination of truth values of the variables.

  9. Truth Table for “AND” • pq is true if p is true and q is true. • pq is false if p is false or q is false.

  10. Truth Table for “OR” • p  q is true if p is true or q is true. • p  q is false if p is false and q is false.

  11. Truth Table for “not” • p is true if p is false. • p is false if p is true.

  12. Example: Truth Table • Truth table for the statement (p)  (q  r ).

  13. Logical Equivalence • Two statements are logically equivalent if they have the same truth values for all combinations of truth values of their variables.

  14. Example: Logical Equivalence • (p  q)  (p q)  (p q)  (p  q)

  15. DeMorgan’s Laws • (pq)  (p)  (q) • (pq)  (p)  (q) • It is not true that “John is short and he is fat”, then it is true that “John is not short or John is not fat”. • If it is not true that x 5 or x 10, then it is true that x > 5 and x < 10.

  16. Tautologies and Contradictions • A tautology is a statement that is logically equivalent to T. • A contradiction is a statement that is logically equivalent to F. • Some tautologies: • pp • pq (pq) • Some contradictions: • pp • pq (pq)

  17. Wrapup • Quiz on Tuesday (Chapter 1) • Homework due Thursday

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