1. Let's start by stating the problem: We want to understand the rules for constructing truth tables in logic. 2. A truth table is a tool used in logic to determine the truth value of a compound statement based on the truth values of its components. 3. The basic truth values are True (T) and False (F). 4. For a statement with $n$ variables, the truth table has $2^n$ rows, representing all possible combinations of truth values. 5. Common logical operators and their truth table rules: - **Negation (NOT, \neg)**: flips the truth value. $$\neg T = F, \quad \neg F = T$$ - **Conjunction (AND, \wedge)**: true only if both operands are true. $$T \wedge T = T, \quad T \wedge F = F, \quad F \wedge T = F, \quad F \wedge F = F$$ - **Disjunction (OR, \vee)**: true if at least one operand is true. $$T \vee T = T, \quad T \vee F = T, \quad F \vee T = T, \quad F \vee F = F$$ - **Implication (IF...THEN, \to)**: false only if the first is true and the second is false. $$T \to T = T, \quad T \to F = F, \quad F \to T = T, \quad F \to F = T$$ - **Biconditional (IF AND ONLY IF, \leftrightarrow)**: true if both operands have the same truth value. $$T \leftrightarrow T = T, \quad T \leftrightarrow F = F, \quad F \leftrightarrow T = F, \quad F \leftrightarrow F = T$$ 6. To build a truth table: - List all variables. - Write all possible combinations of truth values. - Apply the operator rules step-by-step to find the truth value of the compound statement. 7. Example: For $p \wedge \neg q$, with variables $p$ and $q$: - List all combinations: $(T,T), (T,F), (F,T), (F,F)$. - Compute $\neg q$ for each. - Compute $p \wedge \neg q$ using the conjunction rule. This systematic approach helps analyze logical statements clearly and accurately.