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Logical Form and Logical Equivalence

Logical Form and Logical Equivalence. Lecture 2 Section 1.1 Fri, Jan 19, 2007. Statements. A statement is a sentence that is either true or false, but not both. These are statements: It is Wednesday. Discrete Math meets today. These are not statements: Hello. Are you there? Go away!.

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Logical Form and Logical Equivalence

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  1. Logical Form and Logical Equivalence Lecture 2 Section 1.1 Fri, Jan 19, 2007

  2. Statements • A statement is a sentence that is either true or false, but not both. • These are statements: • It is Wednesday. • Discrete Math meets today. • These are not statements: • Hello. • Are you there? • Go away!

  3. Logical Operators • Binary operators • Conjunction – “and”. • Disjunction – “or”. • Unary operator • Negation – “not”. • Other operators • XOR – “exclusive or” • NAND – “not both” • NOR – “neither”

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

  5. Examples • Basic statements • p = “It is Wednesday.” • q = “Discrete Math meets today.” • Compound statements • pq = “It is Wednesday and Discrete Math meets today.” • pq = “ It is Wednesday or Discrete Math meets today.” • p = “It is not Wednesday .”

  6. False Negations • Statement • Everyone likes me. • False negation • Everyone does not like me. • True negation • Someone does not like me.

  7. False Negations • Statement • Someone likes me. • False negation • Someone does not like me. • True negation • No one likes me.

  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 • DeMorgan’s Laws: (pq)  (p)  (q) (pq)  (p)  (q) • If it is not true that i < size && value != array[i] then it is true that…

  16. DeMorgan’s Laws • DeMorgan’s Laws: (pq)  (p)  (q) (pq)  (p)  (q) • If it is not true that i < size && value != array[i] then it is true that i >= size || value == array[i]

  17. DeMorgan’s Laws • If it is not true that x 5 orx  10, then it is true that …

  18. DeMorgan’s Laws • If it is not true that x 5 orx  10, then it is true that x > 5 andx < 10.

  19. Tautologies and Contradictions • A tautology is a statement that is logically equivalent to T. • It is a logical form that is true for all logical values of its variables. • A contradiction is a statement that is logically equivalent to F. • It is a logical form that is false for all logical values of its variables.

  20. Tautologies and Contradictions • Some tautologies: • pp • pq (pq) • Some contradictions: • pp • pq (pq)

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