1. **State the problem:** Given the truth values $A = \text{false}$, $B = \text{true}$, $C = \text{false}$, and $D = \text{true}$, find the truth value of the statement: $$A \wedge \neg B C \wedge \neg C \to \neg D$$ 2. **Interpret the statement:** The statement involves logical AND ($\wedge$), NOT ($\neg$), and implication ($\to$). We need to evaluate each part step-by-step. 3. **Evaluate each component:** - $\neg B = \neg \text{true} = \text{false}$ - $\neg C = \neg \text{false} = \text{true}$ - $\neg D = \neg \text{true} = \text{false}$ 4. **Evaluate $\neg B C$:** Since $\neg B = \text{false}$ and $C = \text{false}$, the conjunction is: $$\text{false} \wedge \text{false} = \text{false}$$ 5. **Evaluate $A \wedge \neg B C$:** Since $A = \text{false}$ and $\neg B C = \text{false}$: $$\text{false} \wedge \text{false} = \text{false}$$ 6. **Evaluate $A \wedge \neg B C \wedge \neg C$:** Since $\neg C = \text{true}$: $$\text{false} \wedge \text{true} = \text{false}$$ 7. **Evaluate the implication:** $$\text{false} \to \neg D = \text{false} \to \text{false}$$ Recall that an implication $P \to Q$ is false only when $P$ is true and $Q$ is false; otherwise, it is true. Here, $P$ is false, so the implication is: $$\text{true}$$ **Final answer:** The truth value of the statement is **true**.