Boolean Algebra A Boolean algebra is a formal system that extends the usual arithmetic of the real numbers to include new algebraic operators called "AND" (...
Boolean Algebra
A Boolean algebra is a formal system that extends the usual arithmetic of the real numbers to include new algebraic operators called "AND" (and), "OR" (or), and "NOT" (not).
The Boolean algebra is represented by a truth table, which is a table that summarizes the truth values of all possible combinations of truth values for the operators.
The truth table for the Boolean operators is as follows:
| Operator | Truth Table |
|---|---|
| AND | True | True |
| OR | True | False |
| NOT | False | True |
Using the truth table, we can determine the truth values of compound statements made with the Boolean operators.
For example, the statement "A and B" is true if both A and B are true, and false otherwise. Similarly, the statement "A or B" is false if both A and B are false, and true otherwise.
The Boolean operators are used in a wide variety of mathematical and logical contexts, including propositional logic, which is a formal system for reasoning about statements. Boolean algebra is also used in computer science to design and implement digital circuits.
Here are some additional examples of statements that can be expressed using the Boolean operators:
A and B: If A is true and B is true, then the statement is true.
A or B: If A is true or B is true, then the statement is true.
NOT A: If A is false, then the statement is true