Boolean Algebra: Rules, Theorems, Properties & Examples

Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson you will discover and use the rules of Boolean algebra to simplify Boolean expressions (statements that are either True or False). The Rules of Boolean algebra are given in a table, and a few examples show how to use them.

Introduction: What's a Boolean?

True or False? That is the question! (Wait, I thought To be or not to be? was the question...) Boolean algebra can help find the answer. Before we delve into Boolean algebra, let's refresh our knowledge in Boolean basics.

Boolean algebra was created by a mathematician George Boole (1815-1864) as an attempt to make the rules of logic precise. In the twentieth century, though, it has since found amazing uses in such fields as digital computing.

George Boole created a mathematical system for logic.
George Boole

As in regular algebra, Boolean algebra uses letters to stand for values and certain symbols to stand for operations on those values. However, there are only two possible values in Boolean algebra: True (1) or False (0). Also, the operations in Boolean algebra are not exactly like the addition, subtraction, multiplication, or division that we are used to seeing in algebra. Instead, we use AND, OR, NOT, and similar operations.

Operations and Truth Tables

Any Boolean variable, p, q, r, etc., may take the value 1 or 0. This value is called its truth value (1 means True, and 0 means False). The letters may stand for specific statements. For example, p could stand for The sky is blue and q could stand for 5 = 6. Clearly p is true (p = 1), and q is false (q = 0).

A truth table may be used to define each operation. In its simplest form, a truth table lists all the possible values for each variable in a separate row. Only one variable means there are only two possible values for it, and so there are two rows. If there are two variables, then there are 4 combinations, hence 4 rows. Three variables will have 8 rows (23). And in general - you guessed it - the number of rows keeps doubling as more variables are introduced, so that if there are n variables, then the table will contain 2n rows!

Truth tables for AND, OR, and NOT

Combining Statements

Expressions or statements may be built by combining Boolean variables with the operations defined above. Then we could ask whether the result is true or false. Following the example above, p AND q would stand for The sky is blue and 5=6. Notice p AND q is False (The sky may be blue, but 5 is definitely not equal to six). If, however, we had considered p OR q (The sky is blue or 5=6), then the overall value of the statement is True (at least one part had to be true when dealing with OR).

Either, or?

One word of caution, the operation OR is not really expressing a choice. For example, if p = The sky is blue and r = 1 + 1 = 2, then p OR q = 1. In other words, p OR q evaluates to True as long as one or both of p or q are True. This is called inclusive or, as opposed to exclusive or.

Although we won't need it for this lesson, there is an operation for exclusive or, written XOR. The truth value of p XOR q will be 0 (False) if both p and q are the same truth value, and 1 (True) if they differ in truth values.

Next Steps

Easy, right? But what if we need to know the truth value of: NOT (NOT p AND q) AND (p OR q) ? Well, this is where the algebra comes in. (By the way, did you know that the word algebra simply means a method or set of rules ?)

The Rules of Boolean Algebra

Let's talk about the rules, starting with an easy one, the Double Negative Rule.

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