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) $a\mathrm{\Delta }b$ (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 $\mathrm{\Delta }$ 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: