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Homederived Boolean operations

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# derived Boolean operations

Recall that a Boolean algebra is an algebraic system $A$ consisting of five operations:

1. two binary operations: the meet $\wedge$ and the join $\vee$,

2. one unary operation: the complementation ${}^{{\prime}}$, and

3. two nullary operations (constants): $0$ and $1$.

From these operations, define the following “derived” operations (on $A$): for $a,b\in A$

1. (subtraction) $a-b:=a\wedge b^{{\prime}}$,

2. (symmetric difference or addition) $a\Delta b$ (or $a+b$)$:=(a-b)\vee(b-a)$,

3. (conditional) $a\to b:=(a-b)^{{\prime}}$,

4. (biconditional) $a\leftrightarrow b:=(a\to b)\wedge(b\to a)$, and

5. (Sheffer stroke) $a|b:=a^{{\prime}}\wedge b^{{\prime}}$.

Notice that the operators $\to$ and $\leftrightarrow$ are dual of $-$ and $\Delta$ 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 $\backslash$ operation | Boolean | Logic | Set |
---|---|---|---|

$\vee$ or $\cup$ | join | logical or | union |

$\wedge$ or $\cap$ | meet | logical and | intersection |

${}^{{\prime}}$ or $\neg$ or ${}^{{\complement}}$ | complement | logical not | complement |

$0$ | bottom element | falsity | empty set |

$1$ | top element | truth | universe |

$-$ or $\setminus$ | subtraction | set difference | |

$\Delta$ or $+$ | symmetric difference | symmetric difference | |

$\to$ | conditional | implication | |

$\leftrightarrow$ | biconditional | logical equivalence | |

$|$ | Sheffer stroke | Sheffer stroke |

Some of the elementary properties of these derived Boolean operators are:

1. $a-0=a$ and $a-a=0-a=a-1=0$,

2. $(A,+,\wedge,0,1)$ is a ring (a Boolean ring),

3. all Boolean operations can be defined in terms of the Sheffer stroke $|$.

The proofs of these properties mimic the proofs for the properties of the corresponding operators found in naive set theory and propositional logic, such as this entry.

## Mathematics Subject Classification

06E05*no label found*03G05

*no label found*06B20

*no label found*03G10

*no label found*

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