# derived Boolean operations

1. 1.
2. 2.

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

3. 3.

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

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

1. 1.

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

2. 2.
3. 3.
4. 4.
5. 5.

Notice that the operators $\to$ and $\leftrightarrow$ are dual of $-$ and $\Delta$ respectively.

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 (http://planetmath.org/SymmetricDifference)
$\to$ conditional implication  $\leftrightarrow$ biconditional logical equivalence
$|$ Sheffer stroke Sheffer stroke

${{{{}\end{center}\inner@par SomeoftheelementarypropertiesofthesederivedBooleanoperatorsare% :\begin{enumerate} \enumerate@itema-0=a and a-a=0-a=a-1=0, \enumerate@item(A,+,\wedge,0,1) is a ring (a Boolean ring), \enumerate@item all Boolean operations can be defined in terms of the Sheffer % stroke |. \end{enumerate}\inner@par Theproofsofthesepropertiesmimictheproofsforthepropertiesofthecorrespondingoperatorsfoundinnaivesettheoryandpropositionallogic% ,suchasthisentry(\texttt{http://planetmath.org/LogicalConnective}).% \begin{flushright}\begin{tabular}[]{|ll|}\hline Title&derived Boolean % operations\\ Canonical name&DerivedBooleanOperations\\ Date of creation&2013-03-22 17:58:49\\ Last modified on&2013-03-22 17:58:49\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&9\\ Author&CWoo (3771)\\ Entry type&Definition\\ Classification&msc 06E05\\ Classification&msc 03G05\\ Classification&msc 06B20\\ Classification&msc 03G10\\ Defines&symmetric difference\\ Defines&conditional\\ Defines&biconditional\\ \hline}\end{tabular}}}\end{flushright}\end{document}$