Biochar is another long term method to storea. Hydrogen b. Oxygen c. Heavy metal d. Carbon

Sure, here are step by step solutions: $A\to B$: - To convert $A$ to $B$, we can use the following equivalence: $A\to B\equiv \neg A\vee B$ - Applying this equivalence, we get: $(\neg p\wedge q)\to (\neg q\wedge r)\equiv \neg (\neg p\wedge q)\vee (\neg q\wedge r)$ - Using De Morgan's laws to simplify the negation, we get: $(\neg p\wedge q)\to (\neg q\wedge r)\equiv (p\vee \neg q)\vee (\neg q\wedge r)$ - Applying the distributive property, we get: $(p\vee \neg q)\vee (\neg q\wedge r)\equiv (p\vee \neg q\vee \neg q)\wedge (p\vee \neg q\vee r)$ - Simplifying the disjunctions, we get: $(p\vee \neg q\vee \neg q)\wedge (p\vee \neg q\vee r)\equiv (p\vee \neg q)\wedge (p\vee \neg q\vee r)$ - This is equivalent to $A\wedge (A\vee C)$, where $A=p\vee \neg q$ and $C=r$. Thus, we have shown that $A\to B$ is equivalent to $A\wedge (A\vee C)$. $B\to C$: - To convert $B$ to $C$, we can use the following equivalence: $B\to C\equiv \neg B\vee C$ - Applying this equivalence, we get: $(\neg p\wedge q)\wedge (p\vee \neg q\vee r)\to r\equiv \neg ((\neg p\wedge q)\wedge (p\vee \neg q\vee r))\vee r$ - Using De Morgan's laws to simplify the negation, we get: $(\neg p\wedge q)\wedge (p\vee \neg q\vee r)\to r\equiv (\neg (\neg p\wedge q)\vee \neg (p\vee \neg q\vee r))\vee r$ - Using De Morgan's laws again, we get: $(\neg p\wedge q)\wedge (p\vee \neg q\vee r)\to r\equiv ((p\vee \neg q)\vee (\neg r))\vee r$ - Simplifying the disjunctions, we get: $(p\vee \neg q\vee \neg r)\vee r\equiv p\vee r$ - This is equivalent to $B\vee C$, where $B=(\neg p\wedge q)\wedge (p\vee \neg q\vee r)$ and $C=r$. Thus, we have shown that $B\to C$ is equivalent to $B\vee C$. $C\to D$: - To convert $C$ to $D$, we can use the following equivalence: $C\to D\equiv \neg C\vee D$ - Applying this equivalence, we get: $p\vee r\to \neg (\neg q\vee \neg r)\equiv \neg (p\vee r)\vee \neg (\neg q\vee \neg r)$ - Using De Morgan's laws to simplify the negation, we get: \neg (p \lor r) \lor \neg (\neg q \lor \neg r) \