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Let's solve the parts of the problem related to the Boolean expression \( X = XYZ + \overline{X} \overline{Y} (\overline{X}Z) \):

### (i) Simplify the expression using Boolean Law

We are given:
\[
X = XYZ + \overline{X} \overline{Y} (\overline{X}Z)
\]

We will simplify this step by step:

1. **Apply the Distributive Law** to the second term \(\overline{X} \overline{Y} (\overline{X}Z)\):

\[
X = XYZ + \overline{X} \overline{Y} \overline{X} Z
\]

Since \(\overline{X} \overline{X} = \overline{X}\), this simplifies to:

\[
X = XYZ + \overline{X} \overline{Y} Z
\]

2. **Factor the common term** \(Z\) from both parts:

\[
X = Z(XY + \overline{X} \overline{Y})
\]

3. Apply **De Morgan's Law** to \(\overline{X} \overline{Y}\), which states \(\overline{X} \overline{Y} = \overline{X + Y}\), giving us:

\[
X = Z(XY + \overline{(X + Y)})
\]

4. The expression inside the parentheses simplifies to:

\[
X = Z(XY + \overline{X + Y})
\]

And this is the simplified form of the Boolean expression.

### (ii) Construct a combinatorial logic circuit

To construct a logic circuit using **inverters**, **OR gates**, and **AND gates**, based on the simplified expression \( X = Z(XY + \overline{(X + Y)}) \):

- **Inverters**: For the \(\overline{(X + Y)}\) term, we need to:
  - Use an OR gate to compute \(X + Y\).
  - Use an inverter (NOT gate) to compute \(\overline{X + Y}\).

- **AND gates**: 
  - One AND gate to compute \(XY\).
  - Another AND gate to combine \(Z\) with the expression \(XY + \overline{(X + Y)}\).

- **OR gate**: 
  - One OR gate to combine \(XY\) and \(\overline{X + Y}\).

So the structure would involve:

1. Two inputs \(X\) and \(Y\) passed to:
   - An AND gate for \(XY\),
   - An OR gate for \(X + Y\) followed by an inverter for \(\overline{X + Y}\).

2. The outputs of these two are then passed through an OR gate, followed by an AND gate with \(Z\) as input.

### (iii) Applications of Boolean Algebra in Information Technology

1. **Digital Circuit Design**: Boolean algebra is used extensively to design and optimize digital circuits, including CPUs, memory units, and logic gates in general.
   
2. **Search Algorithms and Databases**: Boolean logic is applied in database queries, where search conditions use AND, OR, and NOT to filter and retrieve data efficiently.

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Let me know if you'd like more details or further explanations! Here are 5 related questions to deepen your understanding:

1. How does De Morgan's Law simplify Boolean expressions?
2. What is the significance of using NAND gates in Boolean circuits?
3. Can you explain how Boolean algebra simplifies the design of complex logic circuits?
4. What role do Karnaugh maps play in simplifying Boolean functions?
5. How does Boolean algebra relate to binary arithmetic?

**Tip**: Always remember to check for common terms or patterns when simplifying Boolean expressions—they often lead to significant reductions!

Given a Boolean expression, X = XYZ + X̅Y̅(X̅Z):
(i) Simplify the expression using Boolean Law.
(ii) Construct a combinatorial logic circuit using inverters, OR gates, and AND gates for the expression X.
(iii) Give two applications of Boolean algebra in Information Technology.

Learn how to simplify the Boolean expression X = XYZ + X̅Y̅(X̅Z) using Boolean laws like De Morgan's theorem and distributive law. This problem also includes designing a combinatorial logic circuit and discussing two key applications of Boolean algebra in Information Technology.

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