Associative property in the context of Binary operation

In logic, a rule of replacement is a transformation rule that may be applied to only a particular segment of an expression. A logical system may be constructed so that it uses either axioms, rules of inference, or both as transformation rules for logical expressions in the system. Whereas a rule of inference is always applied to a whole logical expression, a rule of replacement may be applied to only a particular segment. Within the context of a logical proof, logically equivalent expressions may replace each other. Rules of replacement are used in propositional logic to manipulate propositions.

Common rules of replacement include de Morgan's laws, commutation, association, distribution, double negation, transposition, material implication, logical equivalence, exportation, and tautology.

In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.

In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like addition and multiplication of integers. They work similarly to integer addition and multiplication, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

More formally, a ring is a set that is endowed with two binary operations (addition and multiplication) such that the ring is an abelian group with respect to addition. The multiplication is associative, is distributive over the addition operation, and has a multiplicative identity element. Some authors apply the term ring to a further generalization, often called a rng, that omits the requirement for a multiplicative identity, and instead call the structure defined above a ring with identity.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ under the operation of composition.

By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition.

In computer science, an expression is a syntactic notation in a programming language that may be evaluated to determine its value of a specific semantic type. It is a combination of one or more numbers, constants, variables, functions, and operators that the programming language interprets (according to its particular rules of precedence and of association) and computes to produce ("to return", in a stateful environment) another value.In simple settings, the resulting value is usually one of various primitive types, such as string, boolean, or numerical (such as integer, floating-point, or complex).

Expressions are often contrasted with statements—syntactic entities that have no value (an instruction).

In computer science, an expression is a syntactic entity in a programming language that may be evaluated to determine its value of a specific semantic type. It is a combination of one or more constants, variables, functions, and operators that the programming language interprets (according to its particular rules of precedence and of association) and computes to produce ("to return", in a stateful environment) another value.In simple settings, the resulting value is usually one of various primitive types, such as string, boolean, or numerical (such as integer, floating-point, or complex).

Expressions are often contrasted with statements—syntactic entities that have no value (an instruction).