1. **State the problem:** Show using absorption properties that the statement $$56 (p \wedge \neg 2) \wedge (p \wedge \neg 2) \vee (2 \wedge \neg r) \leftrightarrow (p \wedge \neg 2)$$ is logically equivalent. 2. **Identify the expression:** The expression is complex and seems to have a typo or misuse of symbols (e.g., 56 and 2 as propositions). Assuming the intended expression is: $$ (p \wedge \neg q) \wedge (p \wedge \neg q) \vee (q \wedge \neg r) \leftrightarrow (p \wedge \neg q) $$ where $q$ replaces 2 for logical clarity. 3. **Recall absorption law:** The absorption law states: $$ A \wedge (A \vee B) \equiv A $$ and $$ A \vee (A \wedge B) \equiv A $$ 4. **Simplify left side:** $$ (p \wedge \neg q) \wedge (p \wedge \neg q) = p \wedge \neg q $$ because $X \wedge X = X$. 5. **Rewrite expression:** $$ (p \wedge \neg q) \vee (q \wedge \neg r) \leftrightarrow (p \wedge \neg q) $$ 6. **Apply absorption:** Since $(p \wedge \neg q)$ is common, and the right side is $(p \wedge \neg q)$, the left side is: $$ (p \wedge \neg q) \vee (q \wedge \neg r) $$ which is equivalent to $(p \wedge \neg q)$ only if $(q \wedge \neg r)$ is absorbed or redundant. 7. **Check absorption:** Using absorption law: $$ A \vee (B) \leftrightarrow A $$ holds if $B$ is absorbed by $A$, which is not generally true here. 8. **Conclusion:** The original expression simplifies to: $$ (p \wedge \neg q) \vee (q \wedge \neg r) \leftrightarrow (p \wedge \neg q) $$ which is not generally a tautology unless $(q \wedge \neg r)$ is false or absorbed. **Final answer:** The expression is logically equivalent to $(p \wedge \neg q)$ only if absorption applies, otherwise not equivalent. Note: The original expression contains numeric literals (56, 2) which are not standard logical variables; assuming $q$ instead of 2 for clarity.