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Difference between revisions of "boolean algebra/functional completeness"
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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 '''}''']]. | + | 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 '''}''']]. Such sets are also called '''universal''' or '''complete''' sets. |
== Examples == | == Examples == | ||
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* 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 Boolean function]] | + | *# [[monotone increasing Boolean function|monotone increasing]] |
| − | *# [[self-dual Boolean function]] | + | *# [[self-dual Boolean function|self-dual]] |
| − | *# [[linear boolean function]] | + | *# [[linear boolean function|linear]] |
| − | *# [[0-preserving Boolean function]] | + | *# [[0-preserving Boolean function|0-preserving]] |
| − | *# [[1-preserving Boolean function]] | + | *# [[1-preserving Boolean function|1-preserving]] |
* Identify functional completeness. | * Identify functional completeness. | ||
Revision as of 13:27, 23 November 2015
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 } and { NOR }. Such sets are also called universal or complete sets.
Examples
The following are some examples of functionally complete sets:
Determining Completeness
| Func | Monotone | Self-dual | Linear | 0-preserving | 1-preserving |
|---|---|---|---|---|---|
| NOT | ✘ | ✔ | ✔ | ✘ | ✘ |
| AND | ✔ | ✘ | ✘ | ✔ | ✔ |
| OR | ✔ | ✘ | ✘ | ✔ | ✔ |
- Given a set of Boolean functions
- Find at least one of each:
- Identify functional completeness.
From the table it can be seen that the following sets are functionally complete: { AND, NOT }, { OR, NOT }, { AND, OR, NOT }.