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Getting Social: Graph Theory and Social Network Analysis
Published in Jesús Rogel-Salazar, Advanced Data Science and Analytics with Python, 2020
It is possible to define a partition of nodes V as the set of subsets of nodes C = {Ci} such that the union of Ci is equal to V and for subsets Ci ∩ Cj = ∅ (with i ≠ j). In other words, the subsets do not overlap and when looking at them together as a whole they regenerate the original set V. We can also define an equivalence relation R on V iff it is reflexive (∀υ ∈ V : υRυ), symmetric (∀u,υ ∈ V : uRυ → υRu) and transitive (∀u, υ, z ∈ V : uRz∧zRu → uRυ). Each equivalence relation determines a partion into equivalence classes [υ] = {u : υRu}; and each partition determines an equivalence relation. Weak and strong connectivities as defined above are equivalence relations, defining weak and strong components. Each equivalence relation determines a partition into equivalence classes, and vice versa.
Relations
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
One of the most important types of relation is an equivalence relation on a set. In this section we define the notion of an equivalence relation and explore the close connection between equivalence relations and partitions of a set. Consider the relation R on the set of living people defined by: x R y if and only if x resides in the same country as y. Assuming each person is resident in only one country, the relation satisfies three obvious properties:
Linear Algebra
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Recall now that an equivalence relation must be reflexive, symmetric, and transitive. Relation R is obviously reflexive since the relative deformation gradient of a configuration with respect to itself equals 1 $ \mathbf 1 $ . Next from the identity A13C=13FT13F=23F12FT23F12F=12FT23C12F $$ A^3_1 \boldsymbol{C}= \ \left(^3_1\boldsymbol{F}\right)^T \ ^3_1\boldsymbol{F}= \left(^3_2\boldsymbol{F}\ ^2_1\boldsymbol{F}\right)^T \ ^3_2\boldsymbol{F}\ ^2_1\boldsymbol{F}= \ \left( ^2_1\boldsymbol{F}\right)^T \ ^3_2\boldsymbol{C}\ ^2_1\boldsymbol{F} $$
Constructing condensed memories in functorial time
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2023
Consider the following construction of a very familiar topological space as the quotient of a profinite set. Let be a compact Hausdorff space. Then a classical and somewhat weird fact is that admits a surjection from a profinite set . One construction is to let be the Stone-Čech compactification of , where is considered as a discrete set. This lets one recover as the quotient of by the equivalence relation . Thus, compact Hausdorff spaces can be thought of as quotients of profinite sets by profinite equivalence relations….This is what happens in the condensed perspective, which only records maps from profinite sets. Scholze (2020)
Signed ring families and signed posets
Published in Optimization Methods and Software, 2021
Kazutoshi Ando, Satoru Fujishige
Now, consider any spanning signed ring family with . Define an equivalence relation ∼ on V as follows. For any we have if and only if for each either or . The equivalence classes associated with the equivalence relation ∼ give a partition of V. By the definition of the equivalence relation we see that each component is divided into two sets and (either but not both possibly empty) such that for each with we have either and , or and .
Triangular norm decompositions through methods using congruence relations
Published in International Journal of General Systems, 2022
Funda Karaçal, Samet Arpacı, Kübra Karacair
We will write with to indicate that a and b are related elements under the binary relation R. We note that any equivalence relation R on L induces a partition of L, whose subsets are the equivalence classes of R. We will denote by L/R the set of all the equivalence classes of R and it is called quotient set. If is an equivalence relation on L, we will denote the equivalence class of an element as