1. The problem is to simplify the logical expression $\sim p \wedge r \vee q \vee \sim r$. 2. Recall that $\vee$ is OR, $\wedge$ is AND, and $\sim$ is NOT. 3. The expression is $(\sim p \wedge r) \vee (q \vee \sim r)$. 4. By associativity and commutativity of OR, rewrite as $(\sim p \wedge r) \vee q \vee \sim r$. 5. Notice that $r \vee \sim r$ is a tautology (always true), but here we have $\sim p \wedge r$ OR $\sim r$. 6. Use distributive property: $(\sim p \wedge r) \vee \sim r = (\sim p \wedge r) \vee \sim r$. 7. This is equivalent to $(\sim p \wedge r) \vee \sim r = (\sim p \wedge r) \vee \sim r$. 8. Using the absorption law: $A \wedge B \vee \sim B = (A \wedge B) \vee \sim B = (A \wedge B) \vee \sim B$ simplifies to $A \vee \sim B$. 9. So $(\sim p \wedge r) \vee \sim r = \sim p \vee \sim r$. 10. Now the whole expression is $\sim p \vee \sim r \vee q$. 11. By commutativity, final simplified form is $q \vee \sim p \vee \sim r$. 12. This means the expression is true if $q$ is true, or $p$ is false, or $r$ is false. Final answer: $q \vee \sim p \vee \sim r$