sequence of bits is a commonly used for such functions. Another common example is the subsets of a set E: to a subset F of E, one can define the indicator function that takes the value 1 on F, and 0 outside F. The most general example is the elements of a Boolean algebra, with all of the foregoing being instances thereof. As with elementary algebra, the purely equational part of the theory may be developed, without considering explicit values for the variables. Operations Basic operations The basic operations of Boolean algebra are conjunction, disjunction, and negation. These Boolean operations are expressed with the corresponding binary operators AND, and OR and the unary operator NOT, collectively referred to as Boolean operators. The basic Boolean operations on variables x and y are defined as follows: {| class="wikitable" style="text-align: center" |- !Logical operation !Operator !Notation !Alternative notations !Definition |- |Conjunction |AND |x∧y |x AND y, Kxy |x∧y = 1 if x = y = 1, x∧y = 0 otherwise |- |Disjunction |OR |x∨y |x OR y, Axy |x∨y = 0 if x = y = 0, x∨y = 1 otherwise |- |Negation |NOT |¬x |NOT x, Nx, x̅, x, !x |¬x = 0 if x = 1, ¬x = 1 if x = 0 |} Alternatively the values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables as follows: If the truth values 0 and 1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic (where x + y uses addition and xy uses multiplication), or by the minimum/maximum functions: One might consider that only negation and one of the two other operations are basic, because of the following identities that allow one to define conjunction in terms of negation and the disjunction, and vice versa (De Morgan's laws): Secondary operations The three Boolean operations described above are referred to as basic, meaning that they can be taken as a basis for other Boolean operations that can be built up from them by composition, the manner in which operations are combined or compounded. Operations composed from the basic operations include the following examples: These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs. {| class="wikitable" style="text-align: center" |+Secondary operations. Table 1 |- ! ! ! ! ! |- !0 !0