Boolean Logic - Truth Tables

This section covers the fundamental concepts of Boolean logic, focusing on constructing and completing truth tables. Truth tables are a crucial tool for understanding and simplifying logical expressions and analyzing digital circuits.

Basic Boolean Operators

The core Boolean operators are:

• AND ($ \land $): The output is true only if both inputs are true.
• OR ($ \lor $): The output is true if at least one input is true.
• NOT ($ eg$): The output is the logical inverse of the input (true becomes false, and vice versa).

Constructing Truth Tables

To construct a truth table, you need to consider all possible combinations of input values for a given number of inputs. The number of rows in the truth table is 2ⁿ, where 'n' is the number of inputs.

Here's the general format:

Input A	Input B	Output (A AND B)
0	0	$ eg ( eg A \land eg B)$
0	1	$ eg ( eg A \land B)$
1	0	$ (A \land eg B)$
1	1	$ (A \land B)$

Example 1: Truth Table for (A AND B)

Let's create a truth table for the expression (A AND B). We have two inputs, A and B, so we need 2² = 4 rows.

A	B	A AND B
0	0	0
0	1	0
1	0	0
1	1	1

Example 2: Truth Table for (A OR B)

Now, let's create a truth table for the expression (A OR B). Again, we have two inputs, A and B, so we need 2² = 4 rows.

A	B	A OR B
0	0	0
0	1	1
1	0	1
1	1	1

Example 3: Truth Table for (NOT A)

Let's create a truth table for the expression (NOT A). We have one input, A, so we need 2¹ = 2 rows.

A	NOT A
0	1
1	0

Completing Truth Tables from Logic Expressions

You can often deduce the truth table for a logic expression directly from the expression itself. For example, consider the expression: $ eg (A \lor B) $.

Using De Morgan's Laws, we can rewrite this as: $ eg A \land eg B $.

The truth table for $ eg A \land eg B $ is:

A	B	NOT A	NOT B	NOT A AND NOT B
0	0	1	1	1
0	1	1	0	0
1	0	0	1	0
1	1	0	0	0

Truth Tables for Circuits

Truth tables are also used to analyze digital circuits. By understanding the logic expression implemented by a circuit, you can construct its truth table.

Practice

Practice constructing truth tables for more complex Boolean expressions. This is a fundamental skill for understanding digital logic.