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Understanding Truth Tables for Negation, Conjunction, and Disjunction

This guide explores the purpose and construction of truth tables for logical operations, specifically negation, conjunction, and disjunction. Truth tables systematically list all possible truth values for statements, utilizing combinations of true (T) and false (F) values. It covers how to create truth tables with two letters (p, q) and three letters (p, q, r), showing the required rows and how truth values alternate. Examples illustrate the connection between logical statements and their truth values, highlighting when certain expressions yield true results.

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Understanding Truth Tables for Negation, Conjunction, and Disjunction

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  1. Section 3.2 Truth Tables for Negation, Conjunction, Disjunction

  2. Purpose of Truth Tables • The purpose of a truth table is to list all the possible truth values of a statement. • We use all the possible combinations of True/False for the letters in the statement: p, q, r

  3. Negation Truth Table • ~p • One letter, truth table has 2 rows

  4. Conjunction Truth Table • Truth value of the conjunction p Λ q • 2 letters = 4 rows

  5. Getting the Truth Table Started • The p rows alternate truth values 2 at a time, since 2 is half of 4 • The q rows alternate truth values 1 at a time, since 1 is half of 2

  6. Conjunction Truth Table • p Λ q is true only when both p and q are true. • Conjunction is usually false.

  7. Disjunction Truth Table

  8. Truth Tables • Example: Truth value of the conjunction p Λ ~q

  9. Truth Tables • Example: Truth value of the conjunction p Λ ~q

  10. Truth Tables • Example: Truth value of the conjunction p Λ ~q

  11. Truth Tables with Three Letters, Comma = Parentheses 3 statements: p, q, r p: You must get up early . q: You must stay up late. r: You must eat a good breakfast. “You must get up early or you must not stay up late, and you must eat a good breakfast.” (p v ~ q) Λ r

  12. Truth Tables with Three Letters • 3 letters require 8 rows in the truth table:

  13. Truth Table for (p v ~ q) Λ r

  14. Truth Table for (p v ~ q) Λ r

  15. Truth Table for (p v ~ q) Λ r

  16. Truth Table for (p v ~ q) Λ r

  17. Truth Tables are not required for known truth values When truth values of statements are known: 12 is less than 13 or 12 is equal to 13. T V F T 6 is an even number and 6 is divisible by 3. T Λ T T

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