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Two objects that are not equal are said to be distinct. [4] Equality is often considered a kind of primitive notion, meaning, its not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothing else"). This makes equality a somewhat slippery ...
two objects being equal but distinct, e.g., two $10 banknotes; two objects being equal but having different representation, e.g., a $1 bill and a $1 coin; two different references to the same object, e.g., two nicknames for the same person; In many modern programming languages, objects and data structures are accessed through references. In ...
As such, for two objects and having descriptors, the similarity is defined as: = = =, where the are non-negative weights and is the similarity between the two objects regarding their -th variable. In spectral clustering , a similarity, or affinity, measure is used to transform data to overcome difficulties related to lack of convexity in the ...
Any two pairs of angles are congruent, [4] which in Euclidean geometry implies that all three angles are congruent: [a] If ∠BAC is equal in measure to ∠B'A'C', and ∠ABC is equal in measure to ∠A'B'C', then this implies that ∠ACB is equal in measure to ∠A'C'B' and the triangles are similar. All the corresponding sides are ...
AAS (angle-angle-side): If two pairs of angles of two triangles are equal in measurement, and a pair of corresponding non-included sides are equal in length, then the triangles are congruent. AAS is equivalent to an ASA condition, by the fact that if any two angles are given, so is the third angle, since their sum should be 180°.
By contrast, the category with a single object and a single morphism is not equivalent to the category with two objects and only two identity morphisms. The two objects in are not isomorphic in that there are no morphisms between them. Thus any functor from to will not be essentially surjective. Consider a category with one object , and two ...
The same distinction holds for comparing objects for equality: most basically there is a difference between identity (same object) and equality (same value), corresponding to shallow equality and (1 level) deep equality of two object references, but then further whether equality means comparing only the fields of the object in question or ...
Similarity: Two objects are similar if one can be transformed into the other by a uniform scaling, together with a sequence of rotations, translations, and/or reflections. Isotopy: Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it.