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

1. two binary operations: the meet and the join ,
2. one unary operation: the complementation , and
3. two nullary operations (constants): 0 and .

From these operations, define the following “derived” operations (on ): for

1. (subtraction) ,
2. (symmetric difference or addition) (or ) ,
3. (conditional) ,
4. (biconditional) , and
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
0	bottom element	falsity	empty set
	top element	truth	universe
or	subtraction		set difference
or	symmetric difference		symmetric difference
	conditional	implication
	biconditional	logical equivalence
	Sheffer stroke	Sheffer stroke

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

1. and ,
2. 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.