Boolean algebra is a mathematical system defined by a binary set. This binary set consists of the numbers 1 and 0. Boolean algebra takes two binary numbers to make one single boolean value. It forms the basis for circuits and algorithms used by computers to produce the information we see on the screen. Boolean algebra focuses on logic; when finding Boolean values, the only options are both 0 and 1, 0 or 1, or neither. These are the only "answers" found when using Boolean algebra. Several logical laws must always be followed when using Boolean algebra.
Boolean algebra is formed in the same way logic is built. It uses the letters A and B -- and so on -- to represent the variables. The symbol "•" represents the word "and," so we could say that A • B is equal to A and B. As in mathematics, it works as a multiplication device, and the product of A • B is AB. The "•" symbol also relates to the number 1 in the binary system. Similarly, the symbol "+" represents "or," so that when we have A + B we have A or B. The answer to A + B is known at the sum, but these two items can't be merged together in the way that they are with the product of "•." The "+" symbol relates to the number 0 in the binary system. Finally, the " ' " symbol shows that the variable is an inverse of itself. Two rules apply to inverses: A + A' = 1, and A • A' = 0.
The Law of Commutativity
The commutative law suggests that when working with mathematical equations that involve simple addition and simple multiplication, it doesn't matter in what order you place the numerals. You always get the same result. You could have A + B, or you could have B + A and they yield the same sum. You can also have A • B or B • A, and they still yield AB.
The Law of Associativity
Similarly, it doesn't matter what order you place groups in when you use addition or multiplication. The combination of (A + B) + C equals the combination (B + C) + A. This holds true if you replace the "+" with a "•."
The Law of Distribution
The distribution law distributes a variable over a group of items. If you have A • (B + C), the product is AB + AC, because you have distributed A to B and A to C, and you must now add them together. This also works if you replace the "•" with a "+," as with A + (B • C) = (A + B) • (A + C).
Solving Boolean Algebra
Truth tables and mapping are used to "solve" Boolean algebra. Because Boolean algebra is binary and because there are a finite number of combinations between the binary set that can be true, you use truth tables to determine the logic and truth of the possible combinations. Truth tables can be created for two or more variables, but all of these variables can only be the numbers 1 or 0. Therefore, you must test each variable that it may possibly be 1 and that it may possibly be 0. Truth tables determine all of the various combinations available. They create possible functions defined by logic. Mapping is a two-dimensional truth table that simplifies these functions even further. These simplified functions reveal the combinations that the binary 0 and 1 can perform and provide the limits on circuitry.
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