Simplify each of the following expressions. Label each identity used. Be mindful that you use the appropriate notation for set algebra or boolean algebra. **Logical and Set-Theoretic Laws**### Associative Laws- Logical: - \((P \lor Q) \lor R = P \lor (Q \lor R)\) - \((P \land Q) \land R = P \land (Q \land R)\)- Set-Theoretic: - \((A \cup B) \cup C = A \cup (B \cup C)\) - \((A \cap B) \cap C = A \cap (B \cap C)\)### Double Negation- Logical: - \(\lnot \lnot P = P\)- Set-Theoretic: - \((A^C)^C = A\)### DeMorgan’s Laws- Logical: - \(\lnot (P \lor Q) = \lnot P \land \lnot Q\) - \(\lnot (P \land Q) = \lnot P \lor \lnot Q\)- Set-Theoretic: - \((A \cup B)^C = A^C \cap B^C\) - \((A \cap B)^C = A^C \cup B^C\)### Distributive Laws- Logical: - \(P \land (Q \lor R) = (P \land Q) \lor (P \land R)\) - \(P \lor (Q \land R) = (P \lor Q) \land (P \lor R)\)- Set-Theoretic: - \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) - \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)### Absorption Laws- Logical: - \(P \lor (P \land Q) = P\) - \(P \land (P \lor Q) = P\)- Set-Theoretic: - \(A \cap (A \cup B) = A\) - \(A \cup (A \cap B) = A\)### Complement Laws- Logical: - \(P \lor \lnot P = T\) - \(P \land \lnot The image contains two logical expressions:i. \[ \left((A \cap A^C) \cup (A \cap (A \cup B))\right)^C \]ii. \[ \neg\left((\neg P \lor \neg Q) \land Q\right) \]No graphs or diagrams are present. These expressions involve set operations and logical operators. - The first expression utilizes set intersection (\(\cap\)), union (\(\cup\)), and complements (\(^C\)).- The second expression involves negation (\(\neg\)), logical OR (\(\lor\)), and logical AND (\(\land\)).

Name: Simplify each of the following expressions. Label each identity used.
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: Simplify each of the following expressions. Label each identity used. Be mindful that you use the appropriate notation for set algebra or boolean algebra. **Logical and Set-Theoretic Laws**### Associative Laws- Logical: - \((P \lor Q) \lor R = P \lo