derived Boolean operations
Recall that a Boolean algebra is an algebraic system consisting of five operations:
-
1.
two binary operations: the meet and the join ,
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2.
one unary operation: the complementation , and
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3.
two nullary operations (constants): and .
From these operations, define the following “derived” operations (on ): for
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1.
(subtraction) ,
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2.
(symmetric difference or addition) (or ),
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3.
(conditional) ,
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4.
(biconditional) , and
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5.
(Sheffer stroke) .
Notice that the operators and are dual of and respectively.
It is evident that these derived operations (and indeed the entire theory of Boolean algebras) owe their existence to those operations and connectives that are found in logic and set theory, as the following table illustrates:
symbol operation | Boolean | Logic | Set |
---|---|---|---|
or | join | logical or | union |
or | meet | logical and | intersection |
or or | complement | logical not | complement |
bottom element | falsity | empty set | |
top element | truth | universe | |
or | subtraction | set difference | |
or | symmetric difference | symmetric difference (http://planetmath.org/SymmetricDifference) | |
conditional | implication | ||
biconditional | logical equivalence | ||
Sheffer stroke | Sheffer stroke |