UNIT 1 COMPUTER SYSTEM BOOLEAN FUNCTION AND ITS CONVERSION

 Q: What is Boolean Function 

A Boolean function is a mathematical expression formed using Boolean variables, logical operators (such as AND, OR, NOT), and parentheses. It describes the relationship between inputs and outputs in a digital logic circuit. Boolean functions are commonly used to represent the behavior of logic gates and circuits.

Q: Rules of Boolean expression conversion

Certainly! Here are some fundamental rules and properties governing Boolean functions:

1. **Commutative Property**: The order of operands in an operation does not affect the result.

   - \( A + B = B + A \)

   - \( A \cdot B = B \cdot A \)

2. **Associative Property**: The grouping of operands in an operation does not affect the result.

   - \( (A + B) + C = A + (B + C) \)

   - \( (A \cdot B) \cdot C = A \cdot (B \cdot C) \)

3. **Distributive Property**: Operations can be distributed across operands.

   - \( A \cdot (B + C) = A \cdot B + A \cdot C \)

   - \( A + (B \cdot C) = (A + B) \cdot (A + C) \)

4. **Identity Element**: There exist identity elements for both addition (OR) and multiplication (AND).

   - \( A + 0 = A \)

   - \( A \cdot 1 = A \)

5. **Complement (Inverse)**: Every element has a complement that, when combined with the element, results in the identity element.

   - \( A + \overline{A} = 1 \)

   - \( A \cdot \overline{A} = 0 \)

6. **Idempotent Law**: Repeating an operation with the same operand does not change the result.

   - \( A + A = A \)

   - \( A \cdot A = A \)

7. **Absorption Law**: Certain expressions can be absorbed into others without changing the result.

   - \( A + (A \cdot B) = A \)

   - \( A \cdot (A + B) = A \)

8. **De Morgan's Laws**: Expressions involving complements can be rewritten using De Morgan's Laws.

   - \( \overline{A + B} = \overline{A} \cdot \overline{B} \)

   - \( \overline{A \cdot B} = \overline{A} + \overline{B} \)

These rules are fundamental to Boolean algebra and are used extensively in the analysis, manipulation, and simplification of Boolean functions and logic circuits. They provide a systematic way to reason about and manipulate digital logic expressions.

How to Write a Boolean Expression to Simplify Circuits

Our first step in simplification must be to write a Boolean expression for this circuit.

This task is easily performed step by step if we start by writing sub-expressions at the output of each gate, corresponding to the respective input signals for each gate.

Remember that OR gates are equivalent to Boolean addition, while AND gates are equivalent to Boolean multiplication.

For example, I’ll write sub-expressions at the outputs of the first three gates:

 

 

. . . then another sub-expression for the next gate:

 

 

Finally, the output (“Q”) is seen to be equal to the expression AB + BC(B + C):

 

 

Now that we have a Boolean expression to work with, we need to apply the rules of Boolean algebra to reduce the expression to its simplest form (simplest defined as requiring the fewest gates to implement):

 

 

The final expression, B(A + C), is much simpler than the original, yet performs the same function.

If you would like to verify this, you may generate a truth table for both expressions and determine Q’s status (the circuits’ output) for all eight logic-state combinations of A, B, and C, for both circuits. The two truth tables should be identical.

 

Generating Schematic Diagrams from Boolean Expressions

Now, we must generate a schematic diagram from this Boolean expression.

To do this, evaluate the expression, following proper mathematical order of operations (multiplication before addition, operations inside parentheses before anything else), and draw gates for each step. 

Remember again that OR gates are equivalent to Boolean addition, while AND gates are equivalent to Boolean multiplication.

In this case, we would begin with the sub-expression “A + C”, which is an OR gate:

 

 

The next step in evaluating the expression “B(A + C)” is to multiply (AND gate) the signal B by the output of the previous gate (A + C):

 

 

Obviously, this circuit is much simpler than the original, having only two logic gates instead of five.

Such component reduction results in higher operating speed (less delay time from input signal transition to output signal transition), less power consumption, less cost, and greater reliability.

 

Multiple Choice Questions

Here are some multiple-choice questions (MCQs) related to Boolean functions:

1. What is the Boolean expression representing the OR operation of variables A and B?

   a) \( A \cdot B \)

   b) \( A + B \)

   c) \( A' + B' \)

   d) \( A' \cdot B' \)

2. What is the result of the Boolean expression \( A \cdot \overline{B} \) when A is true (1) and B is false (0)?

   a) 1

   b) 0

   c) A

   d) B

3. Which Boolean expression represents the complement of variable A?

   a) \( A' \)

   b) \( \overline{A} \)

   c) \( A + 1 \)

   d) \( A \cdot 0 \)

4. What is the result of the Boolean expression \( \overline{A + B} \) when both A and B are false (0)?

   a) 1

   b) 0

   c) A

   d) B

5. Which of the following Boolean expressions represents the NAND operation of variables A and B?

   a) \( A \cdot B \)

   b) \( A + B \)

   c) \( A' + B' \)

   d) \( \overline{A \cdot B} \)

6. What is the simplified form of the Boolean expression \( A + \overline{A} \)?

   a) 1

   b) A

   c) 0

   d) \( A' \)

7. Which Boolean operation is represented by the expression \( A \cdot A \)?

   a) AND

   b) OR

   c) NOT

   d) XOR

8. What is the Boolean expression representing the NOR operation of variables A and B?

   a) \( A \cdot B \)

   b) \( A + B \)

   c) \( A' + B' \)

   d) \( \overline{A + B} \)

9. Which of the following is an example of a Boolean identity?

   a) \( A \cdot B + A' \cdot B + A \cdot B' = A \cdot B + A' \cdot B \)

   b) \( A + B = B + A \)

   c) \( A \cdot A = A \)

   d) \( A \cdot (B + C) = A \cdot B + A \cdot C \)

10. What is the Boolean expression representing the XOR operation of variables A and B?

    a) \( A \cdot B \)

    b) \( A + B \)

    c) \( A' + B' \)

    d) \( A \oplus B \)

Answers:

1. b) \( A + B \)

2. a) 1

3. b) \( \overline{A} \)

4. a) 1

5. d) \( \overline{A \cdot B} \)

6. c) 0

7. a) AND

8. d) \( \overline{A + B} \)

9. b) \( A + B = B + A \)

10. d) \( A \oplus B \)