1. $X = \overline{A.\overline{B}} + \overline{A.(\overline{A}+C)}$

2. $X = \overline{(AB.\overline{C+D}).AB}$

Let's simplify both Boolean expressions step-by-step.

### 1. $ X = \overline{A.\overline{B}} + \overline{A.(\overline{A}+C)} $

#### Step 1: Apply De Morgan's Law
- First, simplify each part separately.

- For the term $ \overline{A.\overline{B}} $, use **De Morgan's law**:
  \[
  \overline{A.\overline{B}} = \overline{A} + B
  \]

- Now for $ \overline{A.(\overline{A}+C)} $, apply **De Morgan's law**:
  \[
  \overline{A.(\overline{A}+C)} = \overline{A} + \overline{\overline{A}+C}
  \]
  Here, $ \overline{\overline{A}+C} $ is simplified further using De Morgan's law:
  \[
  \overline{\overline{A}+C} = A.\overline{C}
  \]
  So the term becomes:
  \[
  \overline{A.(\overline{A}+C)} = \overline{A} + A.\overline{C}
  \]

#### Step 2: Combine the terms
Now substitute these simplified forms back into the original expression:
\[
X = (\overline{A} + B) + (\overline{A} + A.\overline{C})
\]

#### Step 3: Simplify
We can combine $ \overline{A} $ from both terms:
\[
X = \overline{A} + B + A.\overline{C}
\]
This is the simplest form of the Boolean expression.

### Final simplified expression:
\[
X = \overline{A} + B + A.\overline{C}
\]

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### 2. \( X = \overline{(AB.\overline{

$X = \overline{A.\overline{B}} + \overline{A.(\overline{A}+C)}$ and $X = \overline{(AB.\overline{C+D}).AB}$

This tutorial covers the step-by-step simplification of two Boolean algebra expressions using De Morgan's Law and Boolean identities. Learn how to simplify logical expressions like $\overline{A.\overline{B}} + \overline{A.(\overline{A}+C)}$ and $\overline{(AB.\overline{C+D}).AB}$, suitable for students in Grades 10-12.

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