# 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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### Equivalence relation question, R is the relation defined in $\mathbb{Z}$ $xRy$ ('$x$' is in relation with '$y$') if and only if $xy$ > $0$.

The exercise is the following: $R$ is the relation defined in $\mathbb{Z}$, $xRy$ ('$x$' is in relation with '$y$') if and only if $xy > 0$. Analize reflexivity, symmetry, antisymmetry, ...
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### What is meant by relations in set theory? [closed]

Could you explain to me what is meant by relations in the theory of Sets? I understand that relations are sets, therefore the representation of sets can be used to represent relations.tambien que By ...
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### Show that $\sim$ is an equivalence relation.

Suppose $A,B$ sets and nonempty $X\subseteq A$. For $f\in B^A$ define $\hat f = f_{|X} \in B^X$. Define a relation $\sim$ on $B^A$ by $f\sim g \iff \hat f = \hat g$. Show that $\sim$ is an ...
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### Working out if a given relation is reflexive, symmetric or transitive (or all 3?) [duplicate]

On the set of integers, let 𝑥 be related to 𝑦 precisely when x ≠ y Is this Reflexive? Is this Symmetric? Is this Transitive? I'm also wondering if it can be multiple? I assume it can maybe be two ...
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### How to determine the maximal, minimal, greatest, and smallest elements for the subset relation $\subset$?

If I have a set $S=\{\{a, b\}:a, b\in X\}$ and consider the subset relation $\subset$, how can I determine the maximal, minimal, greatest, and smallest elements? Wouldn't all the possible elements ...
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### Show g is not surjective

\begin{align} f: A \rightarrow B, \space g : B \rightarrow A, \end{align} and \begin{align} f \circ g : B \rightarrow B \end{align} is bijective I am not sure how to show that $g$ doesnt have to ...
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### Biconditional Operator in The Logical Expression for Symmetric Relations

"The logical expression for the symmetric relation should contain the biconditional operator <-> instead of implication -> . Because, if you keep ->, then for the case when F -> T ...
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### What is the number of transitive relation containing exactly three ordered pairs?

On a set $A=\{1,2,3,\cdots,n\}$ , what is the number of transitive relations, $t_{n,3}$ that contain exactly $3$ ordered pairs? An example is the relation $\{(1,2),(2,3),(1,3)\}$ I calculated the same ...
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### What is the largest transitive relation that is not the universal relation?

Let $A=\{1,2,3,\cdots, n\}$. Let $T$ denote a transitive relation on $A$ such that $T\neq A\times A$. What is the possibility for the maximum size for $T$? I considered the case $n=3$. I found that ...
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### Is this relation symmetric? I think it is but my professor claims it isn't.

Is the relation $R = \{(a,b),(a,c),(b,a),(b,c),(c,a),(c,b),(d,d)\}$ symmetric? My professor claims that if $(d,d)$ was not included, it would be symmetric, but the inclusion of $(d,d)$ ruins it ...
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### On the definition of the reflexive-transitive hull

The reflexive-transitive closure of a binary relation 𝑅 is the smallest reflexive and transitive binary relation containing 𝑅. Sentences like the above one are simultaneously widespread and ...
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### Writing down the relation in the figure?

So for the first question, I have written down the following: R = {(Tom, Tom)} R = {(Mia, Mia, Bob, Liv)} R = {(Liv, Liv, Tom, Noa)} R = {(Noa, Noa, Tom)} R = {(Gus, Gus, Tom, Liv)} R= {(Kim, Kim, ...
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### Understanding reflexive, symmetric, and antisymmetric relation with an example

Let A be a finite, non-empty, non-singleton set. Let R(A) denote the set of all reflexive relations on A , S(A) denote the set of all symmetric relations on A and T(A) denote the set of all anti-...
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### Is $x^2=y^2$ a symmetric relation?

$R=\{(x,y):x^2=y^2\}$ and I have to determine whether its an equivalence relation. I found that it's reflexive but for the symmetry part I got confused as $x=y$ is sometimes said to be symmetric ...