Logical Form and Logical Equivalence Lecture 2 Section
- Slides: 20
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: l l ¢ It is Wednesday. Discrete Math meets today. These are not statements: l l l Hello. Are you there? Go away!
Logical Operators ¢ Binary operators l l ¢ Unary operator l ¢ Conjunction – “and”. Disjunction – “or”. Negation – “not”. Other operators l l l 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 l l ¢ p = “It is Wednesday. ” q = “Discrete Math meets today. ” Compound statements l l l 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 l ¢ False negation l ¢ Everyone likes me. Everyone does not like me. True negation l Someone does not like me.
False Negations ¢ Statement l ¢ False negation l ¢ Someone likes me. Someone does not like me. True negation l 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 T T F F F T 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 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 F 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 q) T T T F F F F T T
De. Morgan’s Laws: (p q) ( p) ( q) ¢ If it is not true that ¢ i < size && value != array[i] then it is true that…
De. Morgan’s Laws: (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]
De. Morgan’s Laws ¢ If it is not true that x 5 or x 10, then it is true that …
De. Morgan’s 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. l ¢ 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. l It is a logical form that is false for all logical values of its variables.
Tautologies and Contradictions ¢ Some tautologies: p p l p q ( p q) l ¢ Some contradictions: p p l p q ( p q) l
- Logical form and logical equivalence
- Equivalence statement definition
- Aa'bb' pattern nmr
- Dynamic equivalent is
- Dynamic equivalence
- 01:640:244 lecture notes - lecture 15: plat, idah, farad
- Logical equivalence
- Tautology, contradiction or contingency examples
- Logically equivalent
- Logical equivalence in discrete mathematics
- Logical connectives
- Idempotent law truth table
- Equivalence truth table
- 5 examples of proposition
- False equivalence fallacy
- Application of propositional logic
- Logical equivalence
- Present continuous negative and interrogative
- Natural language processing
- Transitive reasoning examples
- Arranging sentences in logical order