# Questions tagged [equivalence-relations]

For questions about relations that are reflexive, symmetric, and transitive. These are relations that model a sense of "equality" between elements of a set. Consider also using the (relation) tag.

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### Describe the distinct equivalence classes resulting from R. [closed]

A relation R is defined on Z by aRb if 5a−b is even. (a) Prove that R is an equivalence relation. (I've already completed this section) (b) Describe the distinct equivalence classes resulting from R. ...
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### I don't understand why this relation is not reflexive

Let $S = \{a, b, c, d\}$ $R_1 : \{(a, a),(d, d)\}$ I don't understand why this relationship is not reflexive. $R_1$ is a subset of $S$x$S$ and every element in $R_1$ is related to itself. the answer ...
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### Troubles with proving R is an equivalence relation [duplicate]

With $A = \mathbb{Z}$ and $B = \mathbb{Z}-\{0\}$, I'm trying to prove that the relation $R$ defined on $A\times B\,$ by $(a,b)R(c,d)\,$ iff $\,ad=bc$, is an equivalent relations. While I do see why ...
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### Faster way to check if two elements belong to the same coset in $\mathbb{Z}_n$?

Q: Let $G=\mathbb{Z}_{25}$ and $H=\langle 13\rangle$ be the subgroup of $G$ generated by $13$. Yes or No: do $24$ and $23$ belong to the same $H$-coset in $G$? No; my reasoning: Know that $2$ ...
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### Relation and Equivalences: Finding the equivalence classes of a relation given

Here is the problem: Let R be the relation on N given by aRb if and only if 5 divides a-b. a. Verify that R is an equivalence relation. b. List the equivalence classes of R as sets. List at least 5 ...
1 vote
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### Nomenclature for reflexive and symmetric relations

What is the common name for binary relations that are reflexive and symmetric (but not necessarily transitive)? In other words: What is the generalization of an equivalence relation where transitivity ...
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### Showing that equivalence in ring of fractions is well defined [duplicate]

I'm reading this answer, in which, one of the step involves showing that: $$s_3(r_1s_2-r_2s_1)=0,\quad s_1(r_2s_3-r_3s_2)=0\implies s_2(r_1s_3-r_3s_1)=0$$ I am utterly confused on how this implication ...
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### Equivalence classes of polynomials under function transformations

Consider the sets of degree $n$ polynomials, $$P_{n} = \big\{ a_n x^n + a_{n-1}x^{n-1} + \cdots a_1 x + a_0\ : a_{n} \neq 0 \big\},$$ and the collection of classical function transformations: ...
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### Is $[\star]$ a limit point of the sequence $\langle [\frac{1}{1}] , [\frac{1}{2}], [\frac{1}{3}], \dots \rangle$?

Suppose $\mathbb{R}$ has the usual topology. Endow $\mathbb{R}$ with an equivalence relation $\sim$ via $x \sim y$ if $x$ and $y$ are irrational or $x \sim y$ if $x = \frac{a}{n}$, $y = \frac{b}{n}$ ...
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### Examples of relations that don't satisfy one of the three properties of an equivalence relation while satisfying the other two?

Just as a question that I have posed to myself: I want to find three relations $(S, \spadesuit)$, $(R, \clubsuit)$ and $(T, \blacksquare)$ for which $(S, \spadesuit)$ doesn't satisfy the reflexive ...
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### Determine the generated equivalence relation

In algebraic topology one often gives generating relations to induce an equivalence relation and I wondered if there is a fast way to determine the equivalence relation. An example is the following. ...
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### Linearisations of a preorder which 'preserve' equivalence classes.

Suppose I have a preorder $\leq$ on a (finite) set $X$ (so $\leq$ is reflexive and transitive). From this, I can construct an equivalence relation by $x\sim y$ if and only if $x\leq y$ and $y\leq x$. ...
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### Continuous map from A to $D^2$ [closed]

I'm proving something and the only thing I do not have yet is that I have to give a continuous function from $A=[-1,1]^2$ to $D^2$ a disk such that $(x,1)$ is being sent to the northpole of the disk ...
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### Propositional Equivalence-Discrete Mathematics

$$p ∧ ((¬p ∨ q) ∧ (¬q ∨ p))$$ is it a contradiction? I have to prove that the statement is a contradiction.But my answer is $p$. I couldn't find out my mistake.
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### Finding a set of representatives to an equivalence relation on sequences [closed]

Define an equivalence relation on the set of infinite real sequences by: a~b iff a-b is bounded. For example, the equivalence class of 0 (the constant sequence 0,0,0,...) is the set of all bounded ...
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### Foundation of mathematical objects modulo isomorphism in ZFC

Suppose we want to define a notion of graph modulo isomorphism. The first thought that comes to mind is to consider the set of all graphs and then quotient it with respect to the equivalence relation ...
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### The function that takes two quotient sets and merges them

I want to know the definition and the well-definedness of the function that takes two quotient sets (disjoint-set data structures), merges them and returns a quotient set. For example, if the function ...
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### Does an equivalence relation on a group play well with the group operation, provided that the equivalence class of the identity is a normal subgroup?

Given an equivalence relation $\sim$ on a group $G$, such that $$a \sim a' \ \text{ and } \ b \sim b' \ \Longrightarrow \ ab \sim a'b' \ ,$$ the equivalence class $[e_G]$ of the identity is a normal ...
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### Proving properties of relations on a power set.

Take $R$ to be the relation defined on $P(\{1, . . . , 100\})$ by $A \sim B$ if and only if $|A \cap B|$ is even. Firstly, am I right to think that for example, $|\{0\}\cap \{1\}| = |\{1\}| = 1$. And ...
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### Equivalence relations, possible typo in textbook answer

9.78. Let $R_1$ and $R_2$ be equivalence relations on a nonempty set A. Prove or disprove the following: If $R_1$ ∩ $R_2$ is symmetric, then so are $R_1$ and $R_2$. The statement is false. Let A = {1,...
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### Can equivalence relations be axiomatized using just one elementary sentence?

Equivalence relations are traditionally axiomatized by the Reflexivity, Symmetry, and Transitivity axioms. However, they can also be axiomatized by Reflexivity and Circularity. (Circularity is this ...
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### Characterization of simple groups in terms of its conjugacy classes [closed]

Recently I have seen a post whose link is the following. I am not able to prove the first statement, namely, "A group $G$ is simple if and only if for any $1 \neq x \in G$, the conjugacy class of ...
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### Can any subset of $\Bbb{N}$ be an equivalence class? [closed]

I am wondering if for any given $x \in P(\Bbb{N})- \{\emptyset\}$ we can find an equivalence relation such that it will have an equivalence class equal to $x$. Extend of this question is whether for ...
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### Reference Request for Axiomatic/Algebraic Big $\mathcal{O}$ and Little $o$

I have seen the formal definitions of big $\mathcal{O}$ and little $o$, and do all right working with them. Still, I have some questions that a good reference might help clear up. In what level of ...
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I am doing an exercise from a number theory textbook for practice and not sure how to approach this problem. Let $S:=(\Bbb R \times \Bbb R)\setminus{(0,0)}$. For $(x,y),(x',y') \in S$ let $(x,y)~\sim ... 3 votes 2 answers 62 views ### Conjugacy classes of an element that are the same Recently I have been studying the transfer homomorphism, and it came to mind that whether conjugacy class of an element with respect to some subgroup is the same as the original group. Namely, if$x \...
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I have a question about members/subsets. Let A be a nonempty set and let B be a subset of the power set $\mathcal{P} ({A})$ of A. Define a relation R from A to B by xRY if x ∈ Y. Give an example of ...