# Questions tagged [relations]

For questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric, etc), a composition of relations, or anything else concerning a relation on a set.

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### What concept of order is introduced in the twentyfold way?

Four of the folds not present in the twelvefold way but introduced in the twentyfold way, rows $5$ and $6$ of the linked table, are defined by the statement that order matters. However, my ...
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### Smallest lattice which is a boolean algebra can contain only one element?

if we have ({1},>=) then 1>=1 therefore reflexive ,anti-symmetric as (1,1) ,transitive . so it is a POSET and 1^1=1 and again 1LUB1=1 so 1 is the complement of itself the set has LUB and GLB for ...
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### Confusion regarding number of ordered pairs for symmetry/asymmetry

My Discrete Mathematics textbook says the following : A relation is symmetric/antisymmetric/transitive even if there’s one pair/triplet that satisfies the condition. This probably means that if I ...
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### On $N \times N$ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$. Show that $R$ is an equivalence relation.

On $N \times N$ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$ a. Show that $R$ is an equivalence relation. My attempt: By definition 6.2.3 $R$ is an equivalence ...
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### A binary relation contained in its square

My colleague (I guess, investigating structure of specific semigroups) is looking for references about binary relations $R\subset X\times X$ such that $R\subset R\circ R$, that is for each $(v,u)\in R$...
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### If R is a symmetric binary relation, what are x and y in set A?

everyone: I've been reading my textbook for discrete math and a few other textbooks on the topic of binary relations, and finding that I'm struggling to understand the definitions. I think a lot can ...
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### Prove, that $M=\{(a,b)\mid a,b \in \mathbb{N_0} \land (a-b) \text{ mod } 4 = 0\}$ is a equivalence relation

Prove, that $M=\{(a,b)\mid a,b \in \mathbb{N_0} \land (a-b) \text{ mod } 4 = 0\}$ is an equivalence-relation. Refl.: $a-a=0 \text{ mod } 4 =0$ Sym.: $\forall x,y \in M: (x,y) \implies (y,x)$ (Not ...
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### Cartesian product and relations of multisets and hybrid sets

I recently encountered multisets and hybrid sets (allowing negative multiplicities), and have a feeling they might be useful for something I'm trying to model. The definitions of both are clear to me, ...
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### What are the ordered pairs when A = {1, 3, 5, 15, 18} and R be defined by xRy if and only if x|y.

I just wanted to confirm I understand correctly: When trying to find the pairs for: A = {1, 3, 5, 15, 18} and R be defined by xRy if and only if x|y First I determine the factors: x|y 1 is a factor of ...
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### If $R = \{(1,2),(1,4),(3,3),(4,1)\}$, then is $(1,2) \in R^2$? (Powers of Relation)

I basically got this: $R^2 =\{(4,4),(1,1),(3,3),(4,2)\}$ But I'm not sure if I should include (1,2) as well since 2 maps to nothing? Thanks
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### Quick question about antisymmetric relationship.

Here we go, It is a really yes or no question. If aRb is a|b then is this antisymmetric? a, b belongs to integers including 0*