| 1 || 0 || 1 |- !1 !0 |0 || 1 || 0 |- !0 !1 |1 || 1 || 0 |- !1 !1 | 1 || 0 || 1 |} Material conditional The first operation, x → y, or Cxy, is called material implication. If x is true, then the result of expression x → y is taken to be that of y (e.g. if x is true and y is false, then x → y is also false). But if x is false, then the value of y can be ignored; however, the operation must return some boolean value and there are only two choices. So by definition, x → y is true when x is false. (relevance logic suggests this definition, by viewing an implication with a false premise as something other than either true or false.) Exclusive OR (XOR) The second operation, x ⊕ y, or Jxy, is called exclusive or (often abbreviated as XOR) to distinguish it from disjunction as the inclusive kind. It excludes the possibility of both x and y being true (e.g. see table): if both are true then result is false. Defined in terms of arithmetic it is addition where mod 2 is 1 + 1 = 0. Logical equivalence The third operation, the complement of exclusive or, is equivalence or Boolean equality: x ≡ y, or Exy, is true just when x and y have the same value. Hence x ⊕ y as its complement can be understood as x ≠ y, being true just when x and y are different. Thus, its counterpart in arithmetic mod 2 is x + y. Equivalence's counterpart in arithmetic mod 2 is x + y + 1. Given two operands, each with two possible values, there are 22 = 4 possible combinations of inputs. Because each output can have two possible values, there are a total of 24 = 16 possible binary Boolean operations. Any such operation or function (as well as any Boolean function with more inputs) can be expressed with the basic operations from above. Hence the basic operations are functionally complete. Laws A law of Boolean algebra is an identity such as between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms