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In mathematics, the associative property [1] is a property of some binary operations that means that rearranging the parentheses in an expression will not change the result. In propositional logic , associativity is a valid rule of replacement for expressions in logical proofs .
The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.
Over a field of characteristic 0, an algebra is power-associative if and only if it satisfies [,,] = and [,,] =, where [,,]:= () is the associator (Albert 1948). Over an infinite field of prime characteristic p > 0 {\displaystyle p>0} there is no finite set of identities that characterizes power-associativity, but there are infinite independent ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
A semigroup is a set S together with a binary operation ⋅ (that is, a function ⋅ : S × S → S) that satisfies the associative property: For all a, b, c ∈ S, the equation (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) holds. More succinctly, a semigroup is an associative magma.
Consider the expression 5^4^3^2, in which ^ is taken to be a right-associative exponentiation operator. A parser reading the tokens from left to right would apply the associativity rule to a branch, because of the right-associativity of ^, in the following way: Term 5 is read. Nonterminal ^ is read. Node: "5^". Term 4 is read. Node: "5^4".
The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states ...
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