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Congruence permits alteration of some properties, such as location and orientation, but leaves others unchanged, like distances and angles. The unchanged properties are called invariants. In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
An example of congruence. The two figures on the left are congruent, while the third is similar to them. The last figure is neither. Congruences alter some properties, such as location and orientation, but leave others unchanged, like distance and angles. The latter sort of properties are called invariants and studying them is the essence of ...
Congruence relation. In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. [1]
Matrix congruence. In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that. where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the Gram matrix ...
An equivalence relation on a set is a binary relation on satisfying the three properties: [1] for all (reflexivity), implies for all (symmetry), if and then for all (transitivity). The equivalence class of an element is defined as [2] The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although ...
Similarity (geometry) In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.
Translation T is a direct isometry: a rigid motion. [1] In mathematics, an isometry (or congruence, or congruent transformation) is a distance -preserving transformation between metric spaces, usually assumed to be bijective. [a] The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning ...
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. [1][2] The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For ...