1. **Problem statement:** Given truth values $p = T$, $q = T$, $r = F$, and $s = T$, find the truth value of each logical statement. 2. **Recall logical operators:** - $\lor$ is OR: true if at least one operand is true. - $\land$ is AND: true if both operands are true. - $\Rightarrow$ is implication: false only if antecedent is true and consequent false. - $\Leftrightarrow$ is biconditional: true if both operands have the same truth value. - $\neg$ is negation: reverses truth value. 3. **Evaluate each statement:** **a.** $(p \lor q) \lor r$ - $p \lor q = T \lor T = T$ - Then $T \lor r = T \lor F = T$ - **Result:** $T$ **b.** $p \lor (q \lor r)$ - $q \lor r = T \lor F = T$ - Then $p \lor T = T \lor T = T$ - **Result:** $T$ **c.** $r \Rightarrow (s \land p)$ - $s \land p = T \land T = T$ - $r \Rightarrow T = F \Rightarrow T = T$ (implication true if antecedent false) - **Result:** $T$ **d.** $p \Rightarrow (r \Rightarrow s)$ - $r \Rightarrow s = F \Rightarrow T = T$ - $p \Rightarrow T = T \Rightarrow T = T$ - **Result:** $T$ **e.** $p \Rightarrow (r \lor s)$ - $r \lor s = F \lor T = T$ - $p \Rightarrow T = T \Rightarrow T = T$ - **Result:** $T$ **f.** $(p \lor r) \Leftrightarrow (r \land \neg s)$ - $p \lor r = T \lor F = T$ - $\neg s = \neg T = F$ - $r \land \neg s = F \land F = F$ - Biconditional $T \Leftrightarrow F = F$ - **Result:** $F$ **g.** $(s \Leftrightarrow p) \Rightarrow (\neg p \lor s)$ - $s \Leftrightarrow p = T \Leftrightarrow T = T$ - $\neg p = \neg T = F$ - $\neg p \lor s = F \lor T = T$ - Implication $T \Rightarrow T = T$ - **Result:** $T$ **h.** $(q \land \neg s) \Rightarrow (p \Leftrightarrow s)$ - $\neg s = \neg T = F$ - $q \land \neg s = T \land F = F$ - $p \Leftrightarrow s = T \Leftrightarrow T = T$ - Implication $F \Rightarrow T = T$ - **Result:** $T$