# Tagged Questions

For reflexive, symmetric and transitive relations. Use it with the tag (relations).

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### Why not just define equivalence relations on objects and morphisms for equivalent categories?

My question is regarding this blog post https://unapologetic.wordpress.com/2007/05/30/equivalence-of-categories/ which I will paraphrase below: The author gives the example of a category $\mathscr{C}$...
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### Proving the Binary Relation is an Equivalence Relation

Let $R$ be a binary relation on a set A and suppose R is symmetric and transitive. Prove the following: If for every $x$ in $A$ there is a $y$ in $A$ such that $x R y$, then $R$ is an equivalence ...
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### Finite group existence of equivalence relation

I was reading about cosets and, given the fact that if $H$ is a non empty subset of a finite group G, we have the following equality $[G:H]|H|=|G|$, I came up with the following question: If the ...
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### Equivalence classes under logical equivalence by 13 valuations

Let L be the set of 5 propositional variables. Under the equivalence relation given by logical equivalence, how many equivalence classes of propositional terms are given the value TRUE by 13 ...
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### Proving a function $\mathbb{Z}_m \to \mathbb{Z}_m,\ [a] \mapsto [a^2 + 3a + 1]$ is well defined

Prove that $\operatorname{poly}\colon \mathbb{Z}_m \to \mathbb{Z}_m$ given by $\operatorname{poly}\colon [a] \mapsto [a^2 + 3a + 1]$ is well defined. This is what I have so far, working in (mod m) ...
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### Why is one relation transitive but the other is not?

From what I have read about a transitive relation is that if xRy and yRz are both true then xRz has to be true. I'm doing some practice problems and I'm a little confused with identifying a ...
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### 51 people enter a raffle with 10 prizes; 7 are pencils and 3 are cars. How may ways are there to give out the prizes?

Assume that no one can win more than one prize. If the prizes were all different, then we have the case that order matters and repeats are allowed, meaning there are $P(51, 10)$ ways of handing out ...
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If $R$ is an equivalence relation, does $R = R^3$ ? I tried for about 40minutes to construct a relation $R$ that is an equivalence relation that when multiplied with itself twice, it will make $R = R^... 1answer 29 views ### Proving that rational equivalence is an equivalence relation on any set. I seek to prove that the rational equivalence relation is an equivalence relation, in that it is reflexive, symmetric, and transitive. The rational equivalence relation is as follows "Two numbers in ... 1answer 25 views ### Show that$[a][\star][b] :=[a \star b]$defines an operation on$G/\sim$. Let$G$be a set equipped with an operation$\star$and an equivalence relation$\sim$. Suppose that$\sim$is compatible with$\star$, i.e., for elements$a$,$a'$,$b$,$b'$of$G$, $$\text{if}\ a \... 1answer 26 views ### Equivalence Relations of n|(x1-x2) How would one prove that if x_1 and x_2 are elements of \mathbb{Z}, then x_1 ~ x_2 <=> n|(x_1 - x_2)? Giving an example, such as n=6 or such would better help me understand the ... 3answers 10 views ### Equivalence Relations and Classes 3 I am studying for a discrete math exam that is tomorrow and the questions on equivalence classes are not making sense to me. Practice Problem: Let \sim be the relation defined on set of pairs (x, ... 3answers 48 views ### What are the equivalence classes for the relation “congruence modulo 5?” I'm still a little mixed up on equivalence classes, so I'm trying to make some connections. I need to be specific of how many there are and what is in each. Here's what I have: Let \mathscr R be ... 1answer 48 views ### Let ∼ be an equivalence relation on a set A, and let a, b ∈ A. Prove that a ∈ [b] iff b ∈ [a]. I have the following proof outline, but I am not sure how to get started proving this. Can anyone point me in the right directon? Proof. Suppose that \sim is an equivalence relation on a set A, ... 1answer 55 views ### Prove that, with vector addition and scalar multiplication well-defined, V/W becomes a vector space over k. Let V be a vector space over a field k and let W be a subspace of V. Prove that, with vector addition and scalar multiplication well-defined, V/W = {v+W | w\in W} becomes a vector space ... 1answer 17 views ### Relation and proving reflexivity The relation R is defined on integers by xRy if and only if x^2y=ymod6. Prove that R is reflexive. So far I have: Let x=y x^2x=xmod6 I don't know how to go from here... because x^2=... 1answer 51 views ### What is the formalism of category theory to express an equivalence relation? Say I have an abstract set X (could be points, functions, functors or whatever). Say I have an equivalence relation R\in X\times X. What would be the category-theory way to express X/R, that ... 1answer 30 views ### Relations and equivalence classes example I'm studying discrete mathematics in my course at university and I'm going through notes on relations, equivalence relations and classes and such. I've come across an example on equivalence classes ... 1answer 57 views ### Equivalence Relations on Products Let G be a group, p a prime dividing |G| and X = \{(x_0,..., x_{p−1})) ∈ G_p:∏_i x_i = 1\}. Let E be the relation defined on X by (x_0, ..., x_{p−1})E(y_0,..., y_{p−1}) if there exists ... 2answers 52 views ### Which of the following are partitions of \mathbb R^2 Is my answer correct? Can someone provide me better explanations for (a) ,(c) and (d)? Which of the following collections of subsets of the plane \Bbb R\times\Bbb R are partitions? (a) \... 0answers 49 views ### proof a code is equivalent to any code with the code word 00..0 of length n Let C and C' be codes over a q-ary alphabet A. We say that C and C' are equivalent if one can be obtained from the other by repeatedly applying the following two operations ... 2answers 37 views ### Show given relation R is equivalence relation on S I will display the exact problem, then my questions. I have searched to the extremes to figure this out and can't: Show that the given relation R is an equivalence relation on set S. S is the ... 1answer 72 views ### Name of and references for the equivalence relation x \sim y :\Longleftrightarrow x^2 = y^2 Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements x,y of a field \mathbb{F}:$$ x \sim y :\Longleftrightarrow ... 1answer 37 views ### Transitivity of a binary relation on the power set I'm studying for a test and there's a question that I've tried and I don't understand: Let$E$be a binary relation on a set$A$; let a binary relation$F$on$\mathcal P (A) \setminus \{\emptyset\...
Let $M = \mathbb{N} \ \mathbb{x} \ \mathbb{N}$. We define the following relation on $M$. Let $(a,b)R(a',b')$ iff $a + b'=a'+b$ We define the set of intergers $\mathbb{Z}$, to be the set of ...