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! Logical Operators Binary operators Conjunction and. Disjunction or. Unary operator Negation
not. Other operators XOR exclusive or NAND not both NOR neither Logical Symbols Statements are represented by letters: p, q, r, etc. means and. means or. means not. 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 . False Negations Statement Everyone
False negation Everyone likes me. does not like me. True negation Someone does not like me. False Negations Statement Someone
False negation Someone likes me. does not like me. True negation No one likes me. 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. 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. p q pq T T T
T F F F T F F F F 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. p q pq T T T T F T
F T T F F F Truth Table for not p is true if p is false. p is false if p is true. p p
T F F T Example: Truth Table Truth table for the statement (p) (q r). p q r (p) (q r ) T
T T T T T F F T F T
F T F F F F T T T F T
F T F F T T F F F T
Logical Equivalence Two statements are logically equivalent if they have the same truth values for all combinations of truth values of their variables. Example: Logical Equivalence (p q) (p q) (p q) (p q) p T q T (p q) (p q) T
(p q) (p q) T T F F F F T F F F
F T T DeMorgans Laws DeMorgans 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 DeMorgans Laws DeMorgans 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] DeMorgans Laws If it is not true that x 5 or x 10, then it is true that DeMorgans Laws If it is not true that x 5 or x 10,
then it is true that x > 5 and x < 10. 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. Tautologies and Contradictions Some tautologies: p p p q (p q) Some contradictions: p p p q (p q)
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