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− | + | A set of logic operations is '''functionally complete''' in [[Boolean algebra]] provided every [[propositional function]] can be expressed entirely in terms of operations in the set - i.e. by combining the various logic operations in a set one could create every truth table. Two notable sets are [[NAND logic|'''{''' NAND '''}''']] and [[NOR logic|'''{''' NOR '''}''']]. | |
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− | A set of logic operations | ||
== Examples == | == Examples == | ||
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== Determining Completeness == | == Determining Completeness == | ||
+ | {| class="wikitable" style="float: right;" | ||
+ | ! Func !! Monotone !! Self-dual !! Linear !! 0-preserving !! 1-preserving | ||
+ | |- | ||
+ | | [[NOT]] || '''✘''' || '''✔''' || '''✔''' || '''✘''' || '''✘''' | ||
+ | |- | ||
+ | | [[AND]] || '''✔''' || '''✘''' || '''✘''' || '''✔''' || '''✔''' | ||
+ | |- | ||
+ | | [[OR]] || '''✔''' || '''✘''' ||'''✘''' || '''✔''' || '''✔''' | ||
+ | |} | ||
* Given a set of Boolean functions | * Given a set of Boolean functions | ||
* Find at least one of each: | * Find at least one of each: | ||
− | *# [[monotone | + | *# [[monotone Boolean function]] |
− | *# [[self-dual Boolean function | + | *# [[self-dual Boolean function]] |
− | *# [[linear boolean function | + | *# [[linear boolean function]] |
− | *# [[0-preserving Boolean function | + | *# [[0-preserving Boolean function]] |
− | *# [[1-preserving Boolean function | + | *# [[1-preserving Boolean function]] |
* Identify functional completeness. | * Identify functional completeness. | ||
From the table it can be seen that the following sets are functionally complete: '''{''' [[AND]], [[NOT]] '''}''', '''{''' [[OR]], [[NOT]] '''}''', '''{''' [[AND]], [[OR]], [[NOT]] '''}'''. | From the table it can be seen that the following sets are functionally complete: '''{''' [[AND]], [[NOT]] '''}''', '''{''' [[OR]], [[NOT]] '''}''', '''{''' [[AND]], [[OR]], [[NOT]] '''}'''. | ||
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