Associative law
Associative law
Any law of the form,
x o (y o z) = (x o y) o z,
where o is a dyadic operation (function) and x o y is the result of applying the operation to x and y (the value of the function for the arguments x and y). Instead of the sign of equality, there may also appear the sign of the biconditional (in the propositional calculus), or of other relations having properties similar to equality in the discipline in question.
In arithmetic there are two associative laws, of addition and of multiplication
x + (y + z) = (x + y) + z.
x X (y X z) = (x X y) X z.
Associative laws of addition and of multiplication hold also in the theory of real numbers, the theory of complex numbers, and various other mathematical disciplines.
In the propositional calculus there are the four following associative laws (two dually related pairs)
[p ? [q ? r]] = [[p ? q] ? r].
[p[qr]] = [[qp]r].
[p +[q + r]] = [[p + q] + r].
[p = [q = r]] = [[p = q] = r].
Also four corresponding laws in the algebra of classes.
As regards exclusive disjunction in the propositional calculus, the caution should be noted that, although p + q is the exclusive disjunction of p and q, and although + obeys an associative law, nevertheless [p + q] + r is not the exclusive disjunction of the three propositions p, q, r — but is rather, “Either all three or one and one only of p, q, r.” — A.C.