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{