Fact-checked by Grok 2 weeks ago

Symmetric relation

In mathematics, a symmetric relation is a binary relation R on a set A such that for all a, b \in A, if aRb, then bRa.^[1] This property ensures that the relation is invariant under the reversal of its ordered pairs, distinguishing it from asymmetric or antisymmetric relations.^[2]^[3] Symmetric relations play a fundamental role in relational mathematics, particularly as one of the three defining properties of an equivalence relation, alongside reflexivity and transitivity.^[4] An equivalence relation on a set induces a partition into equivalence classes, where elements within each class are related symmetrically and transitively.^[4] Common examples include the equality relation on any set, which relates an element only to itself but satisfies symmetry vacuously for distinct pairs, and the relation on integers defined by x \sim y if |x| = |y|, which equates numbers with the same magnitude regardless of sign.^[1]^[2] Another example is the "not equal to" relation on a set, where if a \neq b, then b \neq a.^[5] In graph theory, symmetric relations model undirected graphs, where the adjacency relation between vertices is symmetric: an edge connecting u to v implies a connection from v to u, with no inherent direction.^[6] This correspondence allows symmetric relations to represent bidirectional connections, such as friendships in social networks or physical links in transportation systems.^[7]^[8] Beyond pure mathematics, symmetric relations appear in logic and computer science.^[9]

Definition

Informal Description

A symmetric relation describes a connection between elements where the relationship works equally in both directions, treating the involved elements as interchangeable. If one element stands in this relation to another, the second element necessarily stands in the same relation to the first, ensuring a balanced and reciprocal link without favoring one over the other. This concept is analogous to mutual friendships among people, where if person A considers person B a friend, then person B also considers person A a friend, or to undirected paths between locations, such as roads that allow travel in either direction without distinction.^[2] Binary relations, which pair elements together, often exhibit this symmetry when the pairing is non-directional. The concept of symmetry in binary relations, often referred to as convertibility, emerged in the mid-19th century as part of the development of the calculus of binary relations in mathematics, particularly through the foundational work of Augustus De Morgan, who studied properties like transitivity and symmetry, and later Charles Sanders Peirce and Ernst Schröder.^[10]^[11] It is important to note that such symmetry does not require the related elements to be identical to one another, nor does it demand that every element relates to itself.^[1]

Formal Definition

In mathematics, a binary relation R on a set X is defined as a subset of the Cartesian product X \times X.^[12] The relation R is symmetric if, for all a, b \in X, whenever (a, b) \in R, it follows that (b, a) \in R.^[13] Equivalently, R is symmetric if and only if R = R^{-1}, where R^{-1} denotes the converse (or inverse) relation given by R^{-1} = \{(b, a) \mid (a, b) \in R\}.^[14]^[15] This condition distinguishes symmetric relations from general directed binary relations, which may connect a to b without a reciprocal connection from b to a; symmetry requires bidirectionality for every pair of related elements.^[16]

Examples

In Mathematics

In mathematics, the equality relation on a set X is defined by a \sim b if and only if a = b for all a, b \in X. This relation is symmetric because equality is bidirectional: if a = b, then necessarily b = a, reflecting the inherent symmetry in the definition of equality as an equivalence relation.^[17] Another fundamental example is the congruence relation modulo n, where n is a positive integer. For integers a and b, a \equiv b \pmod{n} if and only if n divides a - b. This relation is symmetric due to the property of divisibility: if n divides a - b, then n also divides b - a = -(a - b), ensuring b \equiv a \pmod{n}. Congruence modulo n plays a central role in number theory, partitioning the integers into equivalence classes that form the ring \mathbb{Z}/n\mathbb{Z}.^[18] In Euclidean geometry, the perpendicularity relation between two lines l_1 and l_2 holds if l_1 \perp l_2, meaning they intersect at a right angle. This relation is symmetric because if l_1 forms a 90-degree angle with l_2, then l_2 necessarily forms the same angle with l_1, as angles at the intersection are complementary pairs. Perpendicularity is essential in defining orthogonal structures, such as coordinate axes in the plane.^[19] In metric spaces, the distance function d defines a symmetric relation via d(a, b) = d(b, a) for all points a, b in the space. This symmetry axiom ensures that the distance between two points is independent of direction, underpinning the geometry of spaces like Euclidean \mathbb{R}^n or more general topologies. For instance, in \mathbb{R}^2 with the Euclidean metric d((x_1, y_1), (x_2, y_2)) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, the equality d(a, b) = d(b, a) follows directly from the squared terms.^[20] For contrast, consider the divides relation on the positive integers, where a \mid b if there exists an integer k such that b = a k. This relation is not symmetric, as $2 \mid 4 holds (with k=2) but $4 \nmid 2 (no such integer k exists). The divides relation thus highlights how symmetry fails in partial orders, unlike the bidirectional examples above.^[21]

In Everyday Contexts

Symmetric relations appear in various everyday scenarios where the connection between two entities is mutual and bidirectional, meaning if one applies to the other, the reverse also holds without implying hierarchy or directionality. A classic example is the familial relation of "sibling of," where if person A is a sibling of person B, then person B is necessarily a sibling of person A, as siblings share the same parental lineage.^[22] This symmetry ignores any potential differences in age or gender, focusing solely on the reciprocal bond. Similarly, the relation "twin of" exemplifies this in cases of identical or fraternal twins, where the mutual identity as twins creates an inherent two-way connection. In social interactions, friendship often functions as a symmetric relation, particularly in mutual acquaintanceships without power imbalances; if individual X considers Y a friend, Y reciprocates the sentiment in a balanced manner.^[23] This bidirectionality underscores the core idea of symmetric relations, where the association is reciprocal by nature. Physical symmetries, such as one object being the "mirror image" of another, also illustrate this concept in basic terms, as the reflection relation holds equally in reverse for paired items like left and right hands. Linguistically, the relation "synonym of" between words demonstrates symmetry, as if word A is a synonym of word B—sharing the same or nearly identical meaning—then word B is likewise a synonym of word A.^[24] In contrast, not all everyday relations are symmetric; for instance, "parent of" is directional and non-symmetric, since if A is the parent of B, B cannot be the parent of A due to the generational hierarchy.^[25] These examples highlight how symmetric relations foster equality and mutuality in common human experiences.

Properties

Basic Properties

A symmetric relation R on a set X exhibits several inherent properties that follow directly from the requirement that R equals its converse R^{-1}. The empty relation \emptyset on X is always symmetric, as it contains no ordered pairs and thus vacuously satisfies the condition that whenever (a, b) \in \emptyset, it follows that (b, a) \in \emptyset.^[13] Similarly, the full relation X \times X is symmetric, since every possible ordered pair (a, b) is included, ensuring that (b, a) is also present for all a, b \in X.^[13] The identity relation I_X = \{(a, a) \mid a \in X\} is symmetric because each pair (a, a) trivially implies itself under the symmetry condition.^[13] Symmetry is preserved under union: if R and S are symmetric relations on X, then R \cup S is symmetric. To see this, suppose (a, b) \in R \cup S; then (a, b) \in R or (a, b) \in S. If (a, b) \in R, symmetry of R gives (b, a) \in R \subseteq R \cup S; similarly if (a, b) \in S.^[13] Additionally, symmetry is preserved under complementation: if R is symmetric on X, then its complement (X \times X) \setminus R is also symmetric. This holds because if (a, b) \notin R, then (b, a) \notin R (since symmetry of R would otherwise imply (a, b) \in R), so both pairs are absent from the complement.^[13]

Derived Properties

A symmetric relation R on a set X is closed under intersection with another symmetric relation S on X; that is, if both R and S are symmetric, then R \cap S is symmetric. To see this, suppose (a, b) \in R \cap S. Then (a, b) \in R, so by symmetry of R, (b, a) \in R; similarly, (b, a) \in S. Thus, (b, a) \in R \cap S.^[26] The restriction of a symmetric relation to a subset is also symmetric. Specifically, if R is symmetric on X and Y \subseteq X, then the restriction R|_Y = \{(a, b) \in R \mid a, b \in Y\} is symmetric on Y. Indeed, if (a, b) \in R|_Y, then (a, b) \in R with a, b \in Y, so (b, a) \in R by symmetry of R, and since b, a \in Y, it follows that (b, a) \in R|_Y.^[27] A relation R on X is symmetric if and only if it equals its inverse R^{-1} = \{(b, a) \in X \times X \mid (a, b) \in R\}. To derive this, assume R is symmetric: for any (a, b) \in R, (b, a) \in R, so every pair in R^{-1} is in R, hence R^{-1} \subseteq R; the reverse inclusion holds by applying the same argument to R^{-1}, which is symmetric if R is. Conversely, if R = R^{-1}, then for (a, b) \in R, (b, a) \in R^{-1} = R.^[28]^[26] The collection of symmetric relations is not closed under composition. For a counterexample, consider the set X = \{a, b, c\} with symmetric relations R = \{(a, b), (b, a)\} and S = \{(b, c), (c, b)\}. The composition R \circ S = \{(a, c)\}, since a R b S c, but (c, a) \notin R \circ S (no intermediate element connects c to a via R and S), so R \circ S is not symmetric.^[26] For a symmetric relation R on a finite set X with |X| = n, the pairs off the diagonal (i.e., excluding possible reflexive loops (x, x)) come in matched pairs (x, y) and (y, x) for x \neq y, so the cardinality of R excluding the diagonal is even. This pairing structure arises because symmetry requires that if (x, y) \in R with x \neq y, then (y, x) \in R, contributing two elements unless x = y. The total number of possible symmetric relations on such a finite set is $2^{n(n+1)/2}, reflecting independent choices for the n diagonal positions and the n(n-1)/2 upper-triangular pairs (each including both directions or neither).^[26]

Relations to Other Relations

With Asymmetric Relations

An asymmetric relation on a set X is defined as a binary relation R \subseteq X \times X such that for all a, b \in X, if aRb, then \neg (bRa).^[29] This property ensures that the relation is strictly one-directional, with no reciprocal pairs. Symmetric relations, by contrast, require that if aRb, then bRa.^[30] Consequently, no non-empty relation can simultaneously satisfy both properties, as the existence of any pair (a, b) \in R with a \neq b would demand bRa (from symmetry) while prohibiting it (from asymmetry), leading to a contradiction.^[30] The only relation that is both symmetric and asymmetric is the empty relation on X. A classic example of an asymmetric relation is the "less than" relation < on the real numbers \mathbb{R}, where if a < b, then it is impossible for b < a.^[30] This contrasts sharply with the symmetric equality relation = on \mathbb{R}, where a = b implies b = a. In the context of tournaments in graph theory, an asymmetric relation corresponds to a complete directed graph (tournament) where exactly one directed edge exists between any pair of distinct vertices, ensuring no 2-cycles (reciprocal edges).^[31] Logically, the asymmetry condition implies that the symmetric part of the relation is empty, meaning R \cap R^{-1} = \emptyset, where R^{-1} is the converse relation consisting of all pairs (b, a) such that aRb.^[29]

With Antisymmetric Relations

An antisymmetric relation on a set X is defined as a binary relation R such that for all a, b \in X, if aRb and bRa, then a = b.^[13] This property ensures that no two distinct elements are related to each other in both directions, effectively prohibiting bidirectional pairs except for self-relations where a = b.^[30] A relation can be both symmetric and antisymmetric only if it is a subset of the equality relation on X, meaning the only pairs it contains are of the form (a, a) for elements a \in X, or the empty relation on a nonempty set.^[30] In particular, the equality relation itself—where a R b if and only if a = b—satisfies both properties, as it is symmetric (since a = b implies b = a) and antisymmetric (since mutual relation forces equality by definition).^[13] To see why this compatibility holds, suppose R is both symmetric and antisymmetric. By symmetry, for any a, b \in X with a R b, it follows that b R a. Then, by antisymmetry, a R b and b R a imply a = b. Thus, the only possible related pairs are those where a = b, reducing R to (a subset of) the equality relation.^[30] The converse is straightforward: any subset of the equality relation inherits both properties vacuously or directly. For an example contrasting these properties, consider the subset relation \subseteq on the power set of a nonempty set, such as \mathcal{P}(\{[1](/page/1)\}). This relation is antisymmetric because if A \subseteq B and B \subseteq A, then A = B, but it is not symmetric, as \emptyset \subseteq \{1\} holds while \{[1](/page/1)\} \not\subseteq \emptyset.^[2] In contrast, the equality relation on \mathcal{P}(\{[1](/page/1)\}) is both symmetric and antisymmetric. In the context of partial orders, which are reflexive, antisymmetric, and transitive relations, antisymmetry specifically prevents the formation of symmetric cycles of length greater than 1, allowing only self-loops (i.e., reflexive pairs where a R a) without introducing inconsistencies in the ordering.^[2] This ensures that distinct elements cannot mutually relate, maintaining the structure's acyclicity beyond trivial loops.^[30]

With Equivalence Relations

An equivalence relation on a set is defined as a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity. Reflexivity requires that every element is related to itself, symmetry ensures that if one element is related to another, the relation holds in both directions, and transitivity means that if one element is related to a second and the second to a third, then the first is related to the third. These properties together allow the relation to partition the set into disjoint equivalence classes, where elements within each class are indistinguishable under the relation.^[32] The symmetry property is essential for the undirected nature of these partitions, guaranteeing that membership in an equivalence class is bidirectional: if a is in the equivalence class of b, then b is in the equivalence class of a. Without symmetry, the relation would not treat the classes as symmetric groupings, potentially leading to directed or hierarchical structures rather than mutual equivalences. For example, the congruence relation modulo n on the integers, where a \equiv b \pmod{n} if n divides a - b, forms an equivalence relation precisely because it combines reflexivity (since n divides $0), symmetry (if n divides a - b, then it divides b - a), and transitivity (if n divides a - b and b - c, then it divides a - c). This symmetry ensures that congruent integers are grouped symmetrically into residue classes.^[32]^[33]^[34] In contrast, relations that are reflexive and transitive but lack symmetry do not qualify as equivalence relations. A standard example is the "divides" relation on the positive integers, where a relates to b if a divides b; this is reflexive (every integer divides itself) and transitive (if a divides b and b divides c, then a divides c), but not symmetric (e.g., $2 divides $4, but $4 does not divide $2). Similarly, the relation \leq on the real numbers is reflexive and transitive but fails symmetry, as $1 \leq 2 holds while $2 \not\leq 1. These examples illustrate how the absence of symmetry prevents the formation of equivalence classes.^[32]^[35] A key characterization is that symmetry combined with transitivity implies reflexivity, but only on the domain of the relation—specifically, for any element a such that there exists some b with (a, b) in the relation, then (a, a) must hold. To see this, if (a, b) is in the relation, symmetry yields (b, a), and transitivity then gives (a, a). However, the relation may not be reflexive on the entire set if there are isolated elements not participating in any pairs, such as in the empty relation on a non-empty set. This partial reflexivity underscores symmetry's role in building toward full equivalence when reflexivity is explicitly added.^[36]^[37]

Applications

In Graph Theory

In graph theory, a symmetric relation R on a finite set of vertices V corresponds directly to an undirected simple graph G = (V, E), where the edge set E consists of all unordered pairs \{a, b\} with a \neq b and aRb.^[7] This mapping excludes self-loops, treating R as irreflexive for standard graph representations, though reflexive symmetric relations can include loops if desired.^[38] The symmetry of R ensures that edges are bidirectional, distinguishing undirected graphs from directed ones where relations may lack this mutuality.^[39] Adjacency in such graphs is defined symmetrically: vertices a and b (with a \neq b) are adjacent if and only if aRb, implying bRa, which precludes directed edges and aligns the relation with the undirected structure.^[40] The adjacency matrix of G, a |V| \times |V| matrix with entries 1 if adjacent and 0 otherwise (diagonal typically 0), is inherently symmetric due to this property.^[41] Representative examples illustrate this correspondence. The complete graph K_n on n vertices features the full symmetric relation minus the diagonal, relating every distinct pair of vertices.^[42] Conversely, the empty graph (edgeless graph) on V corresponds to the empty symmetric relation, with no pairs related.^[38] Key properties of undirected graphs emerge from the symmetric relation. Connectedness is determined by the existence of paths—sequences of adjacent vertices—where symmetry ensures paths are traversable in both directions; two vertices lie in the same connected component if they are related by the transitive closure of R, forming equivalence classes under this reachability relation.^[43] Cliques, as maximal complete subgraphs, correspond to maximal subsets of vertices inducing the full symmetric relation (minus diagonal), where every pair is adjacent.^[42] This framework extends to multigraphs and weighted graphs. In symmetric multigraphs, multiple edges between pairs are allowed while preserving undirected symmetry, generalizing the relation to a multiset of pairs.^[44] For weighted graphs, edges carry real-valued weights, represented by a symmetric weight matrix where the entry w_{ab} = w_{ba} quantifies the relation's strength, such as distance or capacity.^[41] In social network analysis, symmetric relations capture mutual ties such as trust or collaboration, where the connection between individuals is reciprocal and bidirectional, enabling the study of cohesive group dynamics.^[45] These ties form the basis for balance theory, originally formulated by Fritz Heider, which describes how interpersonal sentiments—positive or negative—tend toward structural balance in triadic configurations to minimize cognitive tension and promote stability in social groups.^[46] For instance, mutual trust in professional collaborations reinforces network stability by ensuring that positive relations align consistently across connected actors.^[47] Reciprocity in sociology highlights symmetric exchanges, particularly in practices like gift-giving, where the act of receiving imposes an obligation to return, creating balanced social bonds and alliances as detailed in Marcel Mauss's seminal work on archaic societies.^[48] This principle of balanced reciprocity underscores how symmetric interactions sustain long-term associations by equalizing obligations between parties, differing from one-sided transactions.^[49] A clear example appears in kinship systems, where the relation "cousin of" is inherently symmetric: if person A is the cousin of person B, then person B is the cousin of person A, allowing undirected representations of family trees that emphasize mutual descent without directionality.^[50] Similar to everyday examples like siblings, this symmetry simplifies tracing relational equivalence in bilateral kinship structures.^[51] In psychology, symmetric attitudes in interpersonal relations involve mutual agreement or liking, fostering attraction and harmony through aligned orientations, as modeled in Newcomb's symmetry theory of co-orientation.^[52] This contrasts with asymmetric power dynamics, where one individual's influence dominates, potentially leading to imbalance and tension in relationships.^[53] Empirical studies in social sciences frequently assume symmetry in surveys to model relations as undirected networks for analytical simplicity, yet real-world data often necessitates checks for asymmetry to distinguish directed influences from mutual ones.^[54]

References

[1]
Properties of Binary Relation
Definition(symmetric relation): A relation R on a set A is called symmetric if and only if for any a, and b in A, whenever <a, b> R , <b, a> R .
[2]
[PDF] Chapter 6 Relations
Relations that resemble equality are normally symmetric. For example, the relation X on the integers defined by xXy iff |x| = |y| is symmetric. So is the ...
[3]
[PDF] Relations
Page 3. Relation on a Set. Definition. A relation on a set A is a subset R ⊆ A × A. We often abbreviate the statement (x,y) ∈ R as xRy.
[4]
[PDF] Math 8: Equivalence Relations
A relation ∼ on a set S is called an equivalence relation if ∼ is reflexive, symmetric, and transitive. Examples: 1. Let S = R and define the relation ∼ by (for ...
[5]
[PDF] Relations
Definition (symmetric relation): A relation R on a set A is called symmetric if. ∀ a, b ∈ A (a,b) ∈ R → (b,a) ∈ R. Example 2: • R≠ on A={1,2,3,4}, such that a R ...
[6]
[PDF] GRAPHS Definition 1. An (undirected) graph is a relation E on a set ...
Definition 1. An (undirected) graph is a relation E on a set V that is symmetric and not reflexive. The elements of V are called vertices, and the elements ...<|control11|><|separator|>
[7]
CSCI 2824 Lecture 29: Graph Theory (Basics)
Apr 27, 2014 · An undirected graph is a special kind of directed graph that occurs when the edge relation is symmetric.
[8]
Relationships - CS 10 | Problem solving | Winter 2025
Symmetric relationships lead to undirected graphs. An edge is between A and B. Asymmetrical relationships lead to directed graphs. An edge is from A to B, and ...
[9]
Rel: Properties of Relations - Software Foundations
Symmetric and Antisymmetric Relations. A relation R is symmetric if R a b implies R b a. Definition symmetric {X: Type} (R: relation X) := ∀ a b : X, (R ...
[10]
[PDF] Origins of the Calculus of Binary Relations - Stanford University
The calculus of binary relations was introduced by De Morgan in 1860, and developed by Peirce and Schröder. Tarski, Jónsson, Lyndon, and Monk further developed ...
[11]
[PDF] Binary Relations
Mar 23, 2020 · A binary relation on a set S is a subset of the Cartesian product S × S. This definition is so abstract that you may find it difficult to see ...
[12]
None
### Extracted and Summarized Content
[13]
Relations and Graphs - Discrete Mathematics - An Open Introduction
Using the language of inverse relations, a relation is symmetric if and only if the relation is equal to its inverse. 🔗. Example 2.6.12. Transitive and non ...
[14]
[PDF] Introduction to Relations - FSU Math
For example, if you are asked to show that a relation, R, on A is symmetric, you would suppose that x and y are arbitrary elements of A such that xRy, and then ...
[15]
[PDF] Relations - DSpace@MIT
Symmetry: all edges are bidirectional. Transitivity: Short-circuits—for ... For suppose we have some symmetric relation T with R ⊆ T . Consider (a, b) ...
[16]
[PDF] Naive set theory. - Whitman People
Halmos —Naive Set Theory. John L. Kelley—Introduction to Modern Algebra. R ... in a set is an equivalence relation if it is reflexive, symmetric, and transitive.
[17]
[PDF] Elementary Number Theory: Primes, Congruences, and Secrets
Feb 5, 2014 · This book covers prime numbers, congruences, secret messages, and elliptic curves, aiming to bring the reader closer to this world.
[18]
Relationship of Perpendicularity - Lexique de mathématique - Netmath
Relationship between two lines that form a right angle or between two orthogonal planes. Properties It is symmetric : if l 1 ⊥ l 2 , then l 2 ⊥ l 1.Missing: source | Show results with:source
[19]
[PDF] Metric Spaces - UC Davis Math
A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y ∈ X. The purpose of this chapter is to introduce ...<|separator|>
[20]
[PDF] Discrete Mathematics, Chapters 2 and 9: Sets, Relations and ...
Discrete Mathematics ... Example: Let R = {(a,b) ∈ Z+ × Z+ | a divides b}. This is not an equivalence relation. It is reflexive and transitive, but not symmetric.
[21]
Symmetric Relations - Definition, Formula, Examples - Cuemath
'Is a biological sibling' is a symmetric relation as if one person A is a biological sibling of another person B, then B is also a biological sibling of A.
[22]
Symmetric Relations | Definition, Formula & Examples - BrightChamps
Oct 27, 2025 · Properties of Symmetric Relations · 1. Inverse equals itself: R-1 = R. · 2. Closed under set operations: If R1 and R2 are symmetric, then so are ...
[23]
Super Symmetries: The Magic of Mirror Images | Brilliant Star
Sep 2, 2020 · Most parts of your body have a nearly identical twin that is its “mirror image.” This is called mirror symmetry. People and most animals have mirror symmetry.
[24]
SymmetricRelationship (extJWNL 2.0.5 API)
A symmetric relationship is one whose type is symmetric (its own inverse). An example of a symmetric relationship is synonymy (since, if a is a synonym of b ...
[25]
Asymmetric Relation - GeeksforGeeks
Jul 23, 2025 · The "is a parent of" relationship is asymmetric. If Alice is the parent of Bob, then Bob cannot be the parent of Alice. A symmetric relation is ...
[26]
None
Below is a comprehensive merged summary of symmetric relations from *Discrete Mathematics and Its Applications* by Kenneth Rosen (8th Edition), based on the provided segments. Since the content spans multiple sections and excerpts, I’ll consolidate the information into a detailed narrative with a table for clarity on key properties, proofs, examples, and other details. The response retains all relevant information while avoiding redundancy and ensuring completeness.
[27]
[PDF] Binary Relations 1 Are We Related?
Oct 17, 2005 · A binary relation, R, consists of a set, A, called the domain of R, a set, B, called the codomain of R, and a subset of A × B called the graph ...<|control11|><|separator|>
[28]
Relations
The reflexive symmetric transitive closure of R is the complete relation; given any two sets x and y, we can get from x to ∅ via (R*)-1 and then to y via R*. ...
[29]
[PDF] 1 Relations
R is asymmetric if aRb implies b 6R a for all a, b ∈ A (i.e. R ∩ R−1 = ∅). R is antisymmetric if a R b and b R a imply a = b for all a, b ∈ A (i.e. R ∩ R−1 ⊆ ...
[30]
[PDF] Day 11 - CMU Math
Jun 6, 2012 · Over Z however, we do not have anti-symmetry; −1 | 1 and 1 | −1. So over Z all we have is that the relation is not symmetric (1 | 2 but 2 |/1).Missing: source | Show results with:source
[31]
[PDF] Binary Relations from Tournament Solutions, and Back Again
A tournament may alternatively be defined as a complete, asymmetric directed graph, with X being the set of vertices. Denote the set of all tournaments on X by ...
[32]
7.2: Equivalence Relations
### Summary of Equivalence Relations and Symmetry
[33]
[PDF] Math 127: Equivalence Relations
We claim that ∼ is an equivalence relation: Reflexivity: Since x has the same parity as x, x ∼ x. Symmetry: If x ∼ y, then x and y have the same parity.Missing: converse | Show results with:converse<|control11|><|separator|>
[34]
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/07%3A_Equivalence_Relations/7.03%3A_Equivalence_Classes
[35]
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/07%3A_Relations/7.02%3A_Properties_of_Relations
[36]
[PDF] Untitled
Part (c) A common mistake included a proof that symmetric and transitive implies reflexive. While it is true that symmetric and transitive implies reflexive.
[37]
[PDF] 21-127 Concepts Homework 8: Solutions
case that symmetric and transitive implies reflexive and the rest of the given proof is valid.) 9.11 Wlog we chose door 1. Now the prize has an equal chance ...<|control11|><|separator|>
[38]
[PDF] Graph theory terminology – Undirected graphs
In this course, the term “graph” (without adjective) will always refer to undirected graphs (adjacency is a symmetric relation). (We use “digraph”.
[39]
3.3 Symmetric Relations and Undirected Graphs | Social Networks
Symmetric relations, where a relationship between two nodes is mutual, are represented by undirected graphs, where only one edge exists between them.
[40]
Introduction to Graphs - Tools for Thinking About Programs
A directed graph can always represent an undirected graph by explicitly including symmetric edges. Therefore, we can think of an undirected graph as a ...
[41]
[PDF] Lecture 1: From Data to Graphs, Weighted Graphs and ... - UMD MATH
Feb 4, 2021 · This lecture introduces basic concepts of weighted and undirected graphs, their graph Laplacian, and geometric graphs. Undirected graphs have ...
[42]
14. Some Graph Theory - MIT Mathematics
A complete graph is often called a clique. The size of the largest clique that can be made up of edges and vertices of G is called the clique number of G. The ...
[43]
[PDF] Notes on graph theory
Dec 13, 2010 · – In an undirected graph, connectivity is symmetric, so it's an equiv- alence relation. ∗ Equivalence classes of. ∗. → are called the connected ...<|separator|>
[44]
1.1 Some Basic Definitions about Graphs (Graphs 101)
Multigraphs are graphs that can have multiple (directed) edges between the same pair of nodes, including self loops. For instance, two authors can coauthor a ...
[45]
Introduction to Social Network Methods: Chapter 7 - UCR ITS
If the data are asymmetric or directed, it is possible that actor A can reach actor B, but that actor B cannot reach actor A. With symmetric or undirected data, ...
[46]
[PDF] ATTITUDES AND COGNITIVE ORGANIZATION Fritz Heider (1946)
Attitudes towards persons and causal unit formations influence each other. An attitude towards an event can alter the attitude towards the person who caused ...<|separator|>
[47]
The Structural Balance Theory of Sentiment Networks: Elaboration ...
Structural balance theory attends to a group's network of sentiments and posits that this network alters over time toward particular structural forms.
[48]
RECIPROCITY - jstor
Marcel Mauss in his influential essay The gift (I954) attempted to explain the practice, noted in many societies, of gift exchange. More particularly he ...
[49]
Sociological Implications of Gift Exchange in Multiagent Systems
Gift exchange was found to be an important mean for agents to build long-term associations due to the reciprocal structure of gift giving. The performance ...
[50]
[PDF] Undertanding Human Relations (Kinship Systems) - HAL-SHS
Feb 6, 2016 · ... cousin of a cross-cousin is a sibling, while the cross-cousin of a sibling is a cross-cousin. In other words, Dravidian systems distinguish.
[51]
Kinship Glossary - Anthropology - The University of Alabama
Bilateral (kinship) (c) In the context of cross-cousin marriage, bilateral is used a synonym for symmetrical; i.e., bilateral cross-cousin marriage is the ...
[52]
[PDF] Cognitive Consistency and Mass Communication
In this respect Newcomb's theory is more of a theory of interpersonal attraction than one of attitude change. If we fail to achieve symmetry through ...
[53]
[PDF] Attitude Alignment in Close Relationships
On the basis of principles of balance theory and interdependence theory, this research examined a phenomenon termed attitude alignment, or the tendency of ...
[54]
[PDF] Detecting Cohesive and 2-mode Communities in Directed and ...
Feb 28, 2014 · Another common assumption made by many present community detection methods is that networks are undirected [34, 43]. This implies that ...<|control11|><|separator|>