"Truth tables with conjunctions or disjunctions"

Truth Tables Overview

• A truth table is a mathematical table used to determine the truth or falsity of a logical expression based on all possible truth values of its variables.

Key Concepts

• Variables:

  • Use variables, like p and q, to represent statements.

• Negation:

  • Denoted as $\sim$

  • When $p$ is true, $\sim p$ is false and vice versa.

  • Rule: $\sim p$ has the opposite truth value of $p$.

• Disjunction:

  • Denoted as $p \lor q$ (read as "p or q")

  • It is true when at least one of $p$ or $q$ is true.

  • Rule: $p \lor q$ is false only when both $p$ and $q$ are false.

• Conjunction:

  • Denoted as $p \land q$ (read as "p and q")

  • It is false when either $p$ or $q$ is false.

  • Rule: $p \land q$ is true only when both $p$ and $q$ are true.

Completing a Truth Table

1. Negation for Variable p:

   • Create a column for $\sim p$ and apply the negation rule.

   • Example values:

     • If $p = T$, then $\sim p = F$

     • If $p = F$, then $\sim p = T$

2. Negation for Variable q:

   • Create a column for $\sim q$ in the same manner as above.

3. Disjunction and Conjunction Columns:

   • Combine the truth values for $\sim p$ and $\sim q$ using the disjunction and conjunction rules.

   • Example:

     • For $\sim q \lor \sim p$, evaluate all combinations of truth values for both variables.

Truth Table Structure Example

Conclusion

• The final answer of the truth table focuses on the column derived from combining negations, disjunctions, and conjunctions.

• Important to note the relationships between the truth values for accurate logical conclusions.