Difference between revisions of "User:HelpJazz/workspace"

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-'''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>

Difference between revisions of "User:HelpJazz/workspace"

Revision as of 19:03, January 5, 2008

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