Lecture Notes on Truth Tables

Definitions and Basics

• Truth Values: A statement is either true or false, not both.
• Tautology: A statement that is true in all possible circumstances. In truth tables, if every line is true, it's a tautology.

Conjunctions, Disjunctions, and Conditionals

Conjunction (A and B)

• Notation: A and B
• Truth Value: True only when both statements are true.
  • Example: "I'm a Math 167 student and I'm taking graduate level German" is false unless both are true.

Disjunction (A or B)

• Notation: A or B
• Truth Value: True unless both statements are false.
  • Example: "I am a student at Mississippi State or I am playing in the roller derby" is false if both are false.

Negation (not A or not B)

• Changes the truth value of the statement.
  • Example: If A is true, not A is false and vice versa.

Conditional (A implies B)

• Notation: A → B (A arrow B)
• Truth Value: Think of it as a promise; only false if you break the promise (i.e., true implies false).
  • Example: If "I win the lottery" implies "I give you $1,000," it's only false if I win the lottery and don't give you $1,000.

Building Truth Tables

Components of a Truth Table

• All possible truth values for variables are given.
• Columns for each variable and their negations.
• Columns for conjunctions, disjunctions, and conditionals.

Steps

1. Negations: Change the truth value in the respective column.
2. Conjunction (A and B): Only true when both are true.
3. Disjunction (A or B): False only when both are false.
4. Conditional (A implies B): True unless true implies false.

Examples

• A and B:
  • True only if both A and B are true.
• A or B:
  • True unless both A and B are false.
• A implies B:
  • False only if A is true and B is false.

Application

• In homework, truth tables are given as multiple-choice questions.
• You fill in truth values based on logical operations and definitions.

Practice Problems

• Convert statements to symbolic form using logical operators.
• Determine truth values of complex compound statements.

Key Points

• Understand each logical operator's truth conditions.
• Focus on the relevant columns in truth tables when answering questions.
• Remember the implications rule: it's a promise that is false only if a true statement implies a false one.

Conclusion

• Practice constructing and interpreting truth tables.
• Remember definitions and logic rules for each operation.
• Contact the instructor with questions or for further clarification.