1. **State the problem:** Simplify the Boolean expression $AB + BC + \overline{A}c$. 2. **Recall Boolean algebra rules:** - $A + \overline{A} = 1$ - $A + AB = A$ - Distributive, associative, and commutative properties apply. 3. **Simplify step-by-step:** Start with the expression: $$AB + BC + \overline{A}c$$ Group terms to factor common variables: $$AB + BC + \overline{A}c = B(A + C) + \overline{A}c$$ 4. **Analyze further:** No direct simplification between $B(A + C)$ and $\overline{A}c$ without expanding. Expand $B(A + C)$: $$BA + BC$$ So original expression is: $$BA + BC + \overline{A}c$$ 5. **Look for consensus terms or absorption:** No direct absorption, but consider the expression as is. 6. **Final simplified form:** The expression is already fairly simplified as: $$AB + BC + \overline{A}c$$ No further reduction is possible without additional constraints. **Answer:** $AB + BC + \overline{A}c$