# Tagged Questions

This tag is intended for questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric,...), composition of relations and similar stuff. More-or-less the things about relations taught in the first elementary set theory or discrete math course.

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### How to find inverse of a relation if the inverse isn't a function?

I am trying to find the inverse of the following function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}$ given by $f(a)=\frac{(-1)^a(2a-1)+1}{4}$. I switched $x$ and $y$ and then tried solving for $y$. This ...
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### define a relation $R$ on $S$?

Let S be the set of humans. 1) Define a relation $R$ on $S$ that is reflexive, symmetric, and transitive but not antisymmetric 2) Define a relation $R$ on $S$ that is symmetric and antisymmetric ...
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### Clarification on a reflexive function

$R_1 = \{(a, b) | a ≤ b\}$ is a reflexive function, but I'm confused on why it is. $a≤b$ but doesn't that not necessarily mean that there is an $a$ that will equal $b$? Couldn't all of the $a$ very ...
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### Confusion on sets and relations

I'm confused on how the number of subsets equals the number of relations. If set A = {1, 2} then AxA would be {}, {1}, {2}, {1,2}. I'm confused on how there are $2^{n^2}$ subsets of $A$ x $A$ because ...
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### Understanding Fuzzy Composition operations

There are two common forms of composition operation in Fuzzy Theory: max–min composition max–product composition Let R be a relation that relates elements from ...
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### For a given relation in $\mathbb{N}\times \mathbb{N}$ find the number of elements in it's equivalence class

The whole problem goes like this: We define the relation $R$ in $\mathbb{N}\times \mathbb{N}$ in the following way: $(a,b)R(c,d)$ iff $a-d=c-b$ First find proof the it's a relation of equivalence (...
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### A relation that is both reflexive and irrefelexive

I didn't know that a relation could be both reflexive and irreflexive. However, now I do, I cannot think of an example. So what is an example of a relation on a set that is both reflexive and ...
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### Why is $x \mid y$ over $\Bbb N$ a partial order but not total order?

I understand why $x \mid y$ is an example of a partial order relation over $\Bbb N$. But can someone explain why its not a total order relation? By definition a total order relation on a set $A$ is a ...
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### Why is the Relation R3 Transitive?

Given $A = \{1,2,3,4\}$ in the Relation $\mathcal{R} = \{(1,1),(2,2),(3,3),(4,4)\}$ I understand why $\mathcal{R}$ is Reflexive, Symmetric but why is it also transitive? In my understanding for a ...
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### Equivalence relations and composition, intersections of them

I've been having some trouble with this one, I hope someone get's it. Let S and R be equivalence relations within X. Prove that if R∘S is an equivalence relation, then it is equal to the ...
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### Binary relations between any two sets

I have some doubts regarding relations and binary relations in particular.This is what I understand : 1) The graph $G_R$ of a relation R on X and Y is the subset of X × Y defined by $G_R$ = {(x, y) ∈...