Boolean Logic - IGCSE Computer Science (0478)

Objective: Draw logic circuits based on problem statements, logic expressions or truth tables

This section covers the fundamental concepts of Boolean logic, which are essential for understanding digital circuits and computer science. We will explore basic logic gates, their truth tables, and how to represent and simplify logic expressions. The goal is to be able to translate real-world problems into logical circuits and vice versa.

1. Basic Logic Gates

Logic gates are the building blocks of digital circuits. They perform basic logical operations on one or more binary inputs (0 or 1) to produce a binary output (0 or 1).

• AND (∑): The output is 1 only if both inputs are 1.
• OR (+): The output is 1 if at least one input is 1.
• NOT (~): The output is the inverse of the input (0 becomes 1, and 1 becomes 0).
• XOR (Exclusive OR): The output is 1 if the inputs are different (one is 0 and the other is 1).
• XNOR (Exclusive NOR): The output is 1 if the inputs are the same (both 0 or both 1).

2. Truth Tables

A truth table shows all possible input combinations and their corresponding output for a logic gate.

AND Gate Truth Table:

Input A	Input B	Output (A & B)
0	0	0
0	1	0
1	0	0
1	1	1

OR Gate Truth Table:

Input A	Input B	Output (A + B)
0	0	0
0	1	1
1	0	1
1	1	1

NOT Gate Truth Table:

Input	Output (∑ A)
0	1
1	0

XOR Gate Truth Table:

Input A	Input B	Output (A XOR B)
0	0	0
0	1	1
1	0	1
1	1	0

3. Logic Expressions and Circuit Diagrams

Logic expressions represent combinations of logic gates. These expressions can be simplified using Boolean algebra to create more efficient circuits.

Example: Consider a circuit that outputs 1 only if both inputs A and B are 1. This can be represented by the logic expression: $A \land B$ or $A \cdot B$. The corresponding circuit would consist of an AND gate with inputs A and B.

Example: A circuit that outputs 1 if either A or B is 1. This can be represented by the logic expression: $A \lor B$ or $A + B$. The corresponding circuit would consist of an OR gate with inputs A and B.

Example: A circuit that inverts the input A. This can be represented by the logic expression: $\overline{A}$ or $\neg A$. The corresponding circuit would consist of a NOT gate with input A.

4. Building Logic Circuits

You can create more complex logic circuits by combining basic logic gates.

Example: A circuit that outputs 1 only if A is 1 and B is 0. The logic expression is $A \land \overline{B}$. The circuit would consist of an AND gate with input A and a NOT gate with input B, connected to the AND gate's other input.

5. Simplifying Logic Expressions (Boolean Algebra)

Boolean algebra provides rules for simplifying logic expressions, which can lead to smaller and more efficient circuits. Common simplification rules include:

• Commutative Law: $A \land B = B \land A$ and $A \lor B = B \lor A$
• Associative Law: $(A \land B) \land C = A \land (B \land C)$ and $(A \lor B) \lor C = A \lor (B \lor C)$
• Distributive Law: $A \land (B \lor C) = (A \land B) \lor (A \land C)$ and $A \lor (B \land C) = (A \lor B) \land (A \lor C)$
• Identity Law: $A \land 1 = A$ and $A \lor 0 = A$
• Complement Law: $A \land \overline{A} = 0$ and $A \lor \overline{A} = 1$
• Idempotent Law: $A \land A = A$ and $A \lor A = A$
• Absorption Law: $A \land (A \lor B) = A$ and $A \lor (A \land B) = A$

By applying these laws, you can reduce the number of gates required to implement a logic function.

6. Practice Problems

Consider the following problem statements and draw the corresponding logic circuits:

1. Output is 1 if A is 1 and B is 0.
2. Output is 1 if A is 0 or B is 0.
3. Output is 1 if both A and B are 0.
4. Output is 1 if A is 1 and B is 1.

These problems can be solved by translating the problem statement into a logic expression and then drawing the corresponding circuit diagram using the appropriate logic gates.