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What Is Boolean Algebra?

What is Boolean algebra? Is it as complicated as it sounds? Why is it so important in the day-to-day functioning of the modern world? Keep on reading to find out!

By
Jason Marshall, PhD
4-minute read
Episode #144

AND, OR, and NOT

In the typical math and algebra that we use on a daily basis, the basic arithmetic operations (addition, subtraction, multiplication, and division) are used to manipulate and combine numbers and symbols that represent numbers. Similarly, in the type of math we’re talking about today known as Boolean algebra, three basic operations are used to manipulate bits—which, remember, are the same thing as “true” and “false” values.

The first operation, NOT, is actually rather boring. It simply returns the opposite of the value that you give it. Since there are two possible values you can give (1 or 0), there are two possible outcomes of NOT:

  • NOT 0 = 1 (since 1 is the opposite of 0)

  • NOT 1 = 0 (since 0 is the opposite of 1)

The other two basic operations in Boolean algebra are more interesting. They both take two bits and give you back 0 or 1. The first operation, AND, only returns 1 if both input bits are equal to 1. Otherwise, AND returns 0. So the four possible input values and the values returned by AND are:

  • 0 AND 0 = 0 (both input bits must be 1 to return 1)

  • 1 AND 0 = 0 (…)

  • 0 AND 1 = 0 (…)

  • 1 AND 1 = 1 (since both input bits are 1)

The final operation, OR, returns 1 if either of the input bits (or both) is equal to 1. Otherwise, it returns 0. So the four possible input values and the values returned by OR are:

  • 0 OR 0 = 0 (one or both input bits must be 1 to return 1)

  • 1 OR 0 = 1 (since one or more of the input bits is 1)

  • 0 OR 1 = 1 (…)

  • 1 OR 1 = 1 (…)

I mentioned earlier that you were probably already (unknowingly) familiar with Boolean algebra. So do any of these operations look familiar? If not, I encourage you to go back and check out the Math Dude episodes on the union and intersection of sets and Venn diagrams. Afterwards, ask yourself if there’s a similarity between the union of sets and the OR operation in Boolean algebra. And what about the intersection of sets and the AND operation? Finally, see if you can figure out what the NOT operation might look like on a Venn diagram.

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About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.