# Tagged Questions

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

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### Mapping of equivalence classes of integers modulo $n$

This is an exercise problem from Essentials of Discrete Mathematics (3rd Edition) by David J. Hunter. The problem is as follows: Consider the function $p : \mathbb{Z} \rightarrow \mathbb{Z}/n$ ...
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### Proving that R is a partial Order.

Define the relation $\Bbb R \times \Bbb R$ by $(a,b) \; R$ $(x,y)$ iff $a \le x$ and $b \le y$ , prove that R is a partial ordering for $\Bbb R\times\Bbb R$ . A partial order is if R is reflexive ...
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### Describe the equivalence relation of the following set with the given partition.

Describe the equivalence relation of the following set with the given partition. $\Bbb N$ , $\{\{ 1 \}, \{2,3 \}, \{4,5,6,7\},\{8,9,10,11,12,13,14,15\}....\} .$ What this question has me ...
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### Checking reflexive, symmetric and transitive properties of $\neq$ on $\mathbb{N}$

QS: Indicate if the relation on the given set are reflexive on a given set, which are symmetric, and which are transitive. $\not = \text{on } \Bbb N$ So for this problem I am trying to ...
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### How to prove equivalence of different definitions for compactness?

My workbook considers three different definitions for compactness in logic. It says that it can be shown that these are equivalent, but what would be a strategy to show this? I'm familiar with showing ...
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### Definition of “quotient set”

I searched and searched about quotient set and cannot figure out what it is. At the beginning, I thought it was the same as partitions, but now I'm confused. Can someone show some examples and explain?...
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### Why isn't reflexivity redundant in the definition of equivalence relation?

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity. Doesn't symmetry and transitivity implies reflexivity? Consider the following argument. For any $a$ ...
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### Describe the equivalence classes generated by T

Suppose $S = \{(x,y) \in \mathbb{R}^2\mid y = x + 1\text{ and } 0 < x < 2\}$. Question Describe the equivalence relation T on the real line that is the intersection of all equivalence ...
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### Is $R$ an equivalence relation?

Let $X,Y$ be infinite sets. Define $F$ as $F=\{f:X\rightarrow Y\}$ . We define a binary relation $R$ on $F$: $fRg$ if there is no countable $S\subseteq X$ such that $\forall x\in S \ f(x)\neq g(x)$. ...
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### Trying to prove Equivalency using Boolean Algebra

The question presented was to use boolean algebra to show that XY’Z + X’Y’Z’ + XY’Z’ + X’YZ’ ≡ XYZ’ + XY’Z + XY’Z’ + XYZ’ I've tried using various laws of Boolean algebra, but the answer that I ...
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### Number of distinct equivalence classes of $\mathbb Z_n$ of the “ associate ” equivalence relation

Define an equivalence relation on $\mathbb Z_n$ as : For $a,b \in \mathbb Z_n$ , $a\sim b$ iff $\exists k \in U_n=\mathbb Z_n^{\times}$ such that $a=kb$ (i.e. $a,b$ are related if they are "...
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### Prove that $R\cap S$ is symmetric, transitive, and anti-symmetric.

If you can confirm these are done correctly or offer another way to do so I would greatly appreciate it. Also how would you go about proving $R\cap S$ is reflexive? What assumption if any would be ...
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### Word for equivalence preserving transformations of equations

I am searching for a mathematical term describing an algebraic manipulation of an equation which preserves equivalence. So while adding $2$ to both sides of an equation results in an equivalent ...
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### Proving $g(x) = E_x$.

I'm pretty new to mathematical proof and set theory and I'm having trouble proving this problem. I've started on it, but I don't know where to progress. Problem: Suppose that $f: A \rightarrow B$. ...
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### Equivalence relations and commutative diagrams

Let $\sim$ and $\dot\sim$ be equivanlence relations on the sets X and Y respectively. Suppose $f \in Y^X$ is such that $x \sim y$ implies $f(x) \dot\sim f(y)$ for all $x,y \in X$. Prove that there is ...
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### How can i prove this equivalence relation problem when R isn't defined?

Problem: Prove that if $R\subset A\times A$, and $R\circ R^{-1}\circ R=R$. Then $R^{-1}\circ R$ is a equivalence relation. in $D(R)$ I have nowhere to take the properties i need from... what do I do? ...
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### How can i prove that if $x_0$ is a solution then $[x_0]$ is unique?

$4x\equiv10\pmod6$ I'm not sure what they asking when they say that the equivalence relation of a solution is unique. Also I was able to find the solution -5 with euclids algorithm, is there a more ...
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### Quotient of a graph?

I want to understand quotient of a graph (also called quotient graph), my teacher says that the terms quotient of a graph and a modulo of a graph should be synonyms (even though modulo of a graph ...
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### We Quotient an algebraic structure to generate equivalence classes?

Till now I visualized Quotient groups as a technique to generate equivalence classes whenever needed, as in $\mathbb{Z}/n\mathbb{Z}$. But now I have a feeling of doubt since I haven't seen a book that ...
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### infinite equivalence classes

How would you prove that this relation $$(a,b)R(c,d)$$ if and only if $$a+d=b+c$$ has infinite equivalence classes if it is defined in a set with only non negative integers? I've already proved that ...
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### Proof the existence of a certain bilinear form on the vector space V/T

Let $\mathbf V$ be a vector space (over a field $\mathbf K$) together with a symmetric bilinear form <-,->, and let $\mathbf T$ $\subseteq$ $\mathbf V$ be the orthogonal complement (since I'm not ...
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### Differentiating between equivalence relations and partitions

What is the purpose of defining equivalence relations and partitions separately when they are the same? Are they used for different purposes?
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### Defining a relation to a set

I have a homework question that asks me to define a relation A2 on $Z$ which is an equivalence relation containing three equivalence classes. $$Z = \{a, b, c, d, e\}$$ I understand what equivalence ...
The relation $P$ on $ℝ$ is defined by $xPy$ iff $x^2=y^2.$ (a) Prove that the the relation $P$ is an equivalence relation. (b) Describe the equivalence class of $3$ and $0$. In order to ...