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Line 1: − '''Boolean algebra''' is a division of mathematics which deals with boolean variables, which are variables which can only have two values. Typically these values are 0 and 1, especially within the fields of [[electronics]] or [[computers]], however boolean variables can have other values, such as "true" and "false". Boolean algebra was invented in the nineteenth century by an [[English]] mathematician named George Boole. − ==Three Basic Operations== − The three basic operations of boolean algebra are AND ([[multiplication]]), OR ([[addition]]), and NOT. See the table below for a numerical description. From these functions, the other functions of boolean algebra can be derived. − − {| class="wikitable" align="left" − ! colspan=3 | AND: A•B=C !! !! colspan=3 | OR: A+B=C !! !! colspan=2 | NOT A'=B − |- align="center" − | '''A''' || '''B''' || '''C''' || rowspan=5 | || '''A''' || '''B''' || '''C''' || rowspan=5 | || '''A''' || '''B''' − |- align="center" − | 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 − |- align="center" − | 0 || 1 || 0 || 0 || 1 || 1 || 1 || 0 − |- align="center" − | 1 || 0 || 0 || 1 || 0 || 1 − |- align="center" − | 1 || 1 || 1 || 1 || 0 || 1 − |} − ===AND=== − AND is the boolean equivalent of multiplication. The product of two numbers are non-zero if both numbers are non-zero. In words, the AND function states: "If A AND B are true then C is true". The AND function is commutative, so it results in the same answer no matter what order the values are in. For example, A • B = B • A, and A • (B • C) = (A • B) • C. − − Just as in traditional [[algebra]], [[exponent]]s can be represented as a series of multiplications in boolean algebra, but the results are different. A<sup>n</sup> can be represented as the value A multiplied n times, or A•A•A•A... = C. Boolean multiplication returns a "true" (or a 1) if all values are "true" (or 1), but in this case, all values are always the same. So if A = 0, C = 0, and if A = 1, C = 1. This means that A<sup>n</sup> simply reduces to A! − − ===OR=== − OR is the boolean equivalent of addition. The sum of two positive numbers is non-zero if either number is non-zero. In words, the OR function states: "If A OR B is true then C is true". Like AND, the OR function is commutative. − − ===NOT=== − NOT is a function which is not the equivalent of an algebraic function. It is represented by an apostrophe (A') or an overbar (<math>\bar A</math>). NOT reverses the value of any variable: if A = 0, A' = 1, and if A = 1, A' = 0. − − NOT can be combined with AND and NOT for interesting results. A•A' (A AND NOT A) always equals 0 (false), since no matter the value of A, one of the two values is always 0. Similarly, A+A' (A OR NOT A) always equals 1 (true), since no matter the value of A, one of the two values is always 1. − − ===Order of Operations=== − The [[order of operations]] for boolean algebra is the same as that for traditional algebra, except that there are fewer functions for boolean algebra: parenthesis are evaluated first, followed by multiplication then addition. Bars over multiple variables are treated at the same level as parenthesis. − − ==Derived Functions== − From the three functions mentioned above, other functions can be derived. Some common derivatives are NAND, NOR and XOR. See the table below for a numerical description. − − {| class="wikitable" align="left" − ! colspan=3 | NAND: (A•B)'!! !! colspan=3 | NOR: (A+B) !! !! colspan=3 | XOR (A'B + AB') − |- align="center" − | '''A''' || '''B''' || '''C''' || rowspan=5 | || '''A''' || '''B''' || '''C''' || rowspan=5 | || '''A''' || '''B''' || '''C''' − |- align="center" − | 0 || 0 || 1 || 0 || 0 || 1 || 0 || 0 || 0 − |- align="center" − | 0 || 1 || 1 || 0 || 1 || 0 || 0 || 1 || 1 − |- align="center" − | 1 || 0 || 1 || 1 || 0 || 0 || 1 || 0 || 1 − |- align="center" − | 1 || 1 || 0 || 1 || 0 || 0 || 1 || 1 || 1 − |} − − ===NAND and NOR=== − NAND and NOR are formed by simply performing a NOT on the output of an AND or OR operation, respectively. NAND can be expressed symbolically as A NAND B = (AB)' and NOR can be expressed symbolically as A NOR B = (A+B)'. − − ===XOR=== − XOR (or "exclusive OR") gives a 1 ("true") if either of its inputs are 1, but not both. Symbolically this can be broken down into its basic operations by the expression: A XOR B = A'B + B'A. − − {{clear}} − <nowiki> − [[Category:Mathematics]] − [[Category:Electronics]]</nowiki>
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Revision as of 19:03, January 5, 2008
, 19:03, January 5, 2008and I liked that one too, but I gotsa make way for new