https://file.aiquickdraw.com/mathbot/file/1726533589406ntufbggd.jpg

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:
   \[
   Y = A + \overline{B}CD + BC\overline{D}
   \]

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This simplifies all the given Boolean equations. 

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

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### 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.

Simplify the following Boolean equations using Boolean theorems. Check for correctness using a truth table or K-map:
(a) Y = AC + A'BC
(b) Y = A'B + A'BC + (A' + C)
(c) Y = AB'C'D + ABC' + ABD + A'B'CD + B'CD + BC'D + A

Simplify complex Boolean equations step-by-step using Boolean theorems such as distribution, absorption, and consensus. This exercise covers essential concepts in Boolean algebra and logic simplification, suitable for Grades 10-12. Solutions verified with truth tables or Karnaugh maps.

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