Construct a logic expression
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Subject Notes |
Computer Science
Logic Gates and Logic Circuits - Logic Expression Construction
Logic Gates and Logic Circuits - Logic Expression Construction
Objective
To construct a logic expression representing a given logic circuit or Boolean function.
1. Boolean Algebra Review
Before constructing logic expressions, a strong understanding of Boolean algebra is crucial. Key concepts include:
- AND (Suggested diagram: AND gate): Output is 1 only if both inputs are 1. Symbol: Suggested diagram: AND gate. Boolean operator: Suggested diagram: AND gate. Algebraic form: $x \cdot y$ or $xy$
- OR (Suggested diagram: OR gate): Output is 1 if at least one input is 1. Symbol: Suggested diagram: OR gate. Boolean operator: Suggested diagram: OR gate. Algebraic form: $x + y$ or $x \lor y$
- NOT (Suggested diagram: NOT gate): Inverts the input. Symbol: Suggested diagram: NOT gate. Boolean operator: Suggested diagram: NOT gate. Algebraic form: $\overline{x}$ or $\neg x$
- Implication (Suggested diagram: Implication gate): $x \implies y$ is read as "if x, then y". It's equivalent to $\overline{x} \lor y$. Symbol: Suggested diagram: Implication gate.
- XOR (Exclusive OR) (Suggested diagram: XOR gate): Output is 1 if exactly one input is 1. Symbol: Suggested diagram: XOR gate. Algebraic form: $x \oplus y$ or $x \text{ XOR } y$
- NAND (Suggested diagram: NAND gate): The inverse of AND. Output is 0 only if both inputs are 1. Symbol: Suggested diagram: NAND gate. Algebraic form: $\overline{x \cdot y}$ or $\overline{xy}$
- NOR (Suggested diagram: NOR gate): The inverse of OR. Output is 1 only if both inputs are 0. Symbol: Suggested diagram: NOR gate. Algebraic form: $\overline{x + y}$ or $\overline{x \lor y}$
2. Constructing Logic Expressions
The process of constructing a logic expression involves translating a logic circuit diagram into a Boolean algebra expression. Here's a step-by-step approach:
- Identify the gates and their inputs.
- Apply Boolean algebra laws to simplify the expression. Common laws include:
- Commutative Law: $x \cdot y = y \cdot x$ and $x + y = y + x$
- Associative Law: $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ and $(x + y) + z = x + (y + z)$
- Distributive Law: $x \cdot (y + z) = (x \cdot y) + (x \cdot z)$ and $x + (y \cdot z) = (x + y) \cdot (x + z)$
- Identity Law: $x \cdot 1 = x$ and $x + 0 = x$
- Complement Law: $x \cdot \overline{x} = 0$ and $x + \overline{x} = 1$
- Absorption Law: $x \cdot (x + y) = x$ and $x + (x \cdot y) = x$
- De Morgan's Laws: $\overline{x \cdot y} = \overline{x} + \overline{y}$ and $\overline{x + y} = \overline{x} \cdot \overline{y}$
- Simplify the expression using Boolean algebra identities.
3. Examples
Example 1: Simple AND gate
Circuit: A single AND gate with inputs A and B.
Logic Expression: $A \cdot B$ or $AB$
| A |
B |
A AND B |
| 0 |
0 |
0 |
| 0 |
1 |
0 |
| 1 |
0 |
0 |
| 1 |
1 |
1 |
Example 2: OR gate with multiple inputs
Circuit: A single OR gate with inputs A, B, and C.
Logic Expression: $A + B + C$ or $A \lor B \lor C$
| A |
B |
C |
A OR B OR C |
| 0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
1 |
| 0 |
1 |
0 |
1 |
| 0 |
1 |
1 |
1 |
| 1 |
0 |
0 |
1 |
| 1 |
0 |
1 |
1 |
| 1 |
1 |
0 |
1 |
| 1 |
1 |
1 |
1 |
Example 3: A more complex circuit
Circuit: A circuit with an AND gate, followed by an OR gate.
Logic Expression: $(A \cdot B) + C$ or $(AB) + C$
| A |
B |
C |
A AND B |
(A AND B) + C |
| 0 |
0 |
0 |
0 |
0 |
| 0 |
0 |
1 |
0 |
1 |
| 0 |
1 |
0 |
0 |
0 |
| 0 |
1 |
1 |
0 |
1 |
| 1 |
0 |
0 |
0 |
0 |
| 1 |
0 |
1 |
0 |
1 |
| 1 |
1 |
0 |
1 |
1 |
| 1 |
1 |
1 |
1 |
1 |
4. Truth Tables and Logic Expressions
Truth tables provide a systematic way to determine the logic expression for a given circuit. By analyzing the output of the circuit for all possible input combinations, we can deduce the Boolean expression that represents its behavior.