Let's work through the Boolean equation simplifications step-by-step based on the image you shared:

(a) $Y = AC + \overline{A}BC$

Using Boolean theorems:

1. Apply distribution: $Y = AC + BC\overline{A}$

2. Notice that $AC$ and $BC\overline{A}$ cannot be further simplified using common factors or identities directly. The simplified expression remains:

   $Y = AC + \overline{A}BC$

(b) $Y = \overline{A}B + \overline{A}BC + (\overline{A} + C)$

Let's simplify this step by step:

1. Apply distribution in the second term $\overline{A}BC$: $Y = \overline{A}B + \overline{A}B + C$

2. Notice that $\overline{A}B$ repeats twice, so we can combine like terms: $Y = \overline{A}B + C$

(c) $Y = A\overline{B}\overline{C}D + AB\overline{C} + ABD + \overline{A}\overline{B}CD + \overline{B}CD + BC\overline{D} + A$

This is a more complex expression, so let's simplify it step by step:

1. Look for common factors across terms. Notice that:

   • $A$ is a common factor in $A\overline{B}\overline{C}D$, $AB\overline{C}$, $ABD$, and $A$.
   • $\overline{B}CD$ appears in two places: $\overline{A}\overline{B}CD$ and $\overline{B}CD$.

2. Group the terms based on common factors: $Y = A(\overline{B}\overline{C}D + B\overline{C} + BD + 1) + \overline{B}CD(\overline{A} + 1) + BC\overline{D}$

3. Simplify inside the parentheses:

   • $A(\overline{B}\overline{C}D + B\overline{C} + BD + 1) = A$ (since $x + 1 = 1$ for any Boolean expression $x$).
   • $\overline{B}CD(\overline{A} + 1) = \overline{B}CD$.

4. The simplified result is: $Y = A + \overline{B}CD + BC\overline{D}$

This simplifies all the given Boolean equations.

Would you like to see the truth table or K-map for verification?

Related Questions:

1. What is De Morgan's theorem, and how is it applied to Boolean expressions?
2. How do you construct a truth table for a simplified Boolean equation?
3. Can you explain the process of minimizing a Boolean expression using Karnaugh maps?
4. What are the advantages of simplifying Boolean equations in digital logic design?
5. How does the consensus theorem apply to simplifying Boolean expressions?

Tip:

Always look for common factors or opportunities to apply basic Boolean laws such as absorption, idempotent, and consensus early in simplification to reduce the expression more efficiently.