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{{title|Boolean Algebra}} | {{title|Boolean Algebra}} | ||
− | '''Boolean algebra''' (or less commonly '''symbolic logic''') is a | + | '''Boolean algebra''' (or less commonly '''symbolic logic''') is a branch algebra that deals with only two logic values - [[0]] (corresponding to [[false]]) and [[1]] (corresponding to [[true]]). |
Today, Boolean algebra is the primary mathematical tool used in designing modern digital systems. Switching functions are described using Boolean algebra since they deal with two discrete states - ON and OFF (or 1 and 0). Those functions are in turn implemented via [[transistors]] which act as switches, a natural implementation for representing Boolean algebra operations. Once primitive Boolean operation circuits such as [[NOT]], [[AND]], and [[OR]] [[logic gates|gates]] are implemented, any conceivable system of logic can be implemented using them like Lego pieces. | Today, Boolean algebra is the primary mathematical tool used in designing modern digital systems. Switching functions are described using Boolean algebra since they deal with two discrete states - ON and OFF (or 1 and 0). Those functions are in turn implemented via [[transistors]] which act as switches, a natural implementation for representing Boolean algebra operations. Once primitive Boolean operation circuits such as [[NOT]], [[AND]], and [[OR]] [[logic gates|gates]] are implemented, any conceivable system of logic can be implemented using them like Lego pieces. | ||
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== Operations & Truth tables == | == Operations & Truth tables == | ||
{{main|/operations|truth table|l1=Boolean Operations}} | {{main|/operations|truth table|l1=Boolean Operations}} | ||
− | Boolean algebra has a set of operations that can be performed on Boolean values, those operations are conveniently enough called '''[[binary operation]]s'''. The three common Boolean operators are '''[[conjunction|AND]]''', '''[[disjunction|OR]]''', and '''[[negation|NOT]]'''. Understanding those operators can better be done by examining their behavior via tool called a truth table. '''[[truth tables]]''' is a table that lists all possible input values and their respective output values. Truth tables can be said to be the unique signature of a specific Boolean function. Truth tables are an excellent way of seeing the relationships between input values and given Boolean expressions. While there may be many ways to realize or construct a Boolean function to represent a specific relation, they all share the very same truth table | + | Boolean algebra has a set of operations that can be performed on Boolean values, those operations are conveniently enough called '''[[binary operation]]s'''. The three common Boolean operators are '''[[conjunction|AND]]''', '''[[disjunction|OR]]''', and '''[[negation|NOT]]'''. Understanding those operators can better be done by examining their behavior via tool called a truth table. '''[[truth tables]]''' is a table that lists all possible input values and their respective output values. Truth tables can be said to be the unique signature of a specific Boolean function. Truth tables are an excellent way of seeing the relationships between input values and given Boolean expressions. While there may be many ways to realize or construct a Boolean function to represent a specific relation, they all share the very same truth table. |
=== AND operator === | === AND operator === | ||
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\end{align} | \end{align} | ||
</math> | </math> | ||
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==Order of operations== | ==Order of operations== | ||
{{main|/order of operations|l1=Order of Operations}} | {{main|/order of operations|l1=Order of Operations}} | ||
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===functional completeness=== | ===functional completeness=== | ||
− | {{main|/functional completeness| | + | {{main|/functional completeness|Functional Completeness}} |
A set of Boolean functions <math>\mathcal{F}</math> is said to be universal or complete iif it's [[/0-preserving function|0-preserving]], [[/1-preserving function|1-preserving]], [[/self-dual function|Self-dual]], [[/monotonically increasing function|monotonically increasing]], and [[/linear function|linear]]. A complete set can form any other Boolean expression by combining the functions in the set. | A set of Boolean functions <math>\mathcal{F}</math> is said to be universal or complete iif it's [[/0-preserving function|0-preserving]], [[/1-preserving function|1-preserving]], [[/self-dual function|Self-dual]], [[/monotonically increasing function|monotonically increasing]], and [[/linear function|linear]]. A complete set can form any other Boolean expression by combining the functions in the set. | ||
== Complementary Function== | == Complementary Function== | ||
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{{empty section}} | {{empty section}} | ||
== See also == | == See also == | ||
* [[Karnaugh Map]] | * [[Karnaugh Map]] | ||
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