0
votes
1answer
42 views

Show that R is an equivalence relation on X for x, y in X iff f(x) = f(y)

$f:X→Y$ $x,y ∈ X,xRy$ iff $f(x) = f(y)$ Show that R is an equivalence relation on X. Also when $X = Y = \mathbb{R}$ and $f: \mathbb{R} \to \mathbb{R} $ with $x \mapsto x^2$ for all $x∈R$ find the ...
0
votes
1answer
46 views

What is the cardinality?

Let $A=\left\{1,2,\cdots,10\right\}$ Let $f,g:A\to A$. Consider the equivalence relation $$ fRg \iff \exists h:A\to A. f=h\circ g$$ where $h$ is invertible. Now, let $g(x)=5$: Why is $\left| ...
2
votes
1answer
58 views

Proving that a relation is an equivalence relation

I am having difficulties proving the relation IS an equivalence relation. Let $f: X\longrightarrow Y$ be a function from a set $X$ onto a set $Y$. Let $R$ be the subset of $X \times X$ consisting ...
1
vote
0answers
19 views

prove the following equivalence $\ln(-\frac{l-x+\sqrt{r^2+(l-x)}}{l+x-\sqrt{r^2+(l+x)}})={}$ArcSinh$(\frac{l+x}{r})+{}$ArcSinh$(\frac{l-x}{r})$

Hi guys i'm trying to prove the following equivalence but i'm having some problems: $$\ln\left(-\frac{\ell-x+\sqrt{r^2+(\ell-x)^{2}}}{\ell+x-\sqrt{r^2+(\ell+x)^{2}}}\right) = \operatorname{ArcSinh} ...
0
votes
2answers
126 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
1
vote
2answers
47 views

Properties of bijections

If a bijection exists between set A={a1, a2, ...} and set B={b1, b2, ...} such that a1 maps to b1 and a2 maps to b2, etc., does this mean if we find a relationship R between a1 and b1 (i.e. f(b1) is ...
0
votes
3answers
33 views

Why is max($\frac{2}{||w||}$)= min($\frac{1}{2}$)($||w||^2$)?

I was watching a video on machine learning. The instructor says that maximizing ($\frac{2}{||w||}$)is difficult (why?) so instead we prefer to minimize $\frac{1}{2}||w||^2$. $w$ is a vector. How are ...
0
votes
1answer
73 views

Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
0
votes
0answers
123 views

About Kernel and the coimage of a function

Introduction I was serching for a concept of "equivalence relations" induced by an arbitrary function in a "natural" way and I found the concept of Kernel. But I'm not sure that I understand it and ...
1
vote
1answer
25 views

Show that f is a symmetric relation on A

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here: Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ ...
8
votes
3answers
177 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
2
votes
2answers
69 views

Equivalence Classes of a Relation Given as a Set of Ordered Pairs

Question: The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. A = {a, b, c, d} R = {(a, a), (b, b), (b, d), (c, c), (d, b), (d, d)} My work: So when ...
3
votes
1answer
71 views

Surjections and equivalence relations

(a) Let $f: A \to B$ be a surjective function. We define $a_1 \sim a_2$ if $f(a_1)=f(a_2)$. Prove that $\sim$ is an equivalence relation. Reflexivity: This comes for free. If $a_1 \sim a_1$, ...
0
votes
2answers
278 views

A Well-Defined Bijection on An Equivalence Class

DATA: Let $f:X\rightarrow Y$ be a surjective function. Define a relation $\sim$ on $X$ by $$a\sim b~\iff~f(a)=f(b).$$ Let $S=X/{\sim}$, namely let $S$ be the set of equivalence classes of elements ...
-2
votes
2answers
122 views

$h:\mathbb{R}_{/\sim}\rightarrow \mathbb{R}^2$: A Bijection from a Quotient Space to the Unit Circle (Geometrically Considered)

NOTE: This is not a duplicate. Define a relation $\sim$ on $\mathbb{R}$ as follows: for any $a,b \in \mathbb{R}$, $$a\sim b \iff a-b\in \mathbb{Z}.$$ Let $S=\mathbb{R}_{/\sim}$. That is, $S$ is ...
0
votes
2answers
112 views

Revisted$_2$: Are doubling and squaring well-defined on $\mathbb{R}/\mathbb{Z}$?

Define a relation $\sim$ on $\mathbb{R}$ as follows: for any $a,b \in \mathbb{R}$, $$a\sim b \iff a-b\in \mathbb{Z}.$$ Let $S=\mathbb{R}/{\sim}$. That is, $S$ is the set of equivalence classes of ...
0
votes
3answers
70 views

An Equivalence Relation: Introspection into a Particular Well-Defined Quotient

DATA: Let $f:\mathbb{Z}\setminus \{0\}\rightarrow \mathbb{N}$ be a function defined by $$f(n) = \{k~:~n=2^km,~m\in \cal{O}\},$$ where $\cal{O}$ is the set of odd integers. Let ...
0
votes
1answer
78 views

Relation $R$: $R\circ R \subseteq R \implies R$ is transitive

Let $R$ be a relation on $X$, a set. If $R\circ R\subseteq R$, then is $R$ transitive?
3
votes
2answers
57 views

Functional relations : Trouble seeing transitivity

Given the following domain: $\;\{1,2,3,4\}$ And the following relation: $$\{(1,1),(1,3),(1,5),(2,2),(2,4),(3,1), (3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(5,5)\}$$ It states that this is an equivalence ...
4
votes
2answers
205 views

Counting the number of functions

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is ...
1
vote
1answer
103 views

Quick equivalence class clarification question

A quick clarification question, what is an equivalence class of a function? For example if you have an identity function on all integers $I_{Z}$, what would $[I_{Z}]$ = ? I know that when you have a ...
5
votes
2answers
497 views

Sets, functions and relations problem

Yes this is a homework problem but I have attempted to solve it and my work is below, also this is my first question here so I'm sorry for any mistakes: Question: Context: Let $A$ and $B$ be subsets ...
0
votes
2answers
310 views

Function and equivalence relations question

Let A and B be subsets of the set Z of all integers, and let F denote the set of all functions f : A to B. Define a relation R on F by: for any f,g element of F, fRg if and only if f - g is a ...
2
votes
2answers
839 views

Proving two functions are equal

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is ...
1
vote
2answers
464 views

Find the number of distinct equivalence classes $[f]$ of $R$.

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Define a relation $R$ on $\mathscr F$ by : for any $f,g ...
0
votes
1answer
58 views

Number of Equivalence Classes

Let $M=\{1,2,\ldots,20\}$ and define a function $f:M\to \mathbf{Z}$ by $f(x)=\min(x,3)$. Define an equivalence relation on M by letting two element $m$ and $n$ be equivalent if $f(m)=f(n)$. 1) How ...
2
votes
3answers
130 views

Is $f:\mathbb{Z}_{30}\longrightarrow\mathbb{Z}_{30}$ defined by $f([a])=[7a]$ well defined?

To tell the truth, I'm not even sure what this means. The professor gave an example saying that $\mathbb{Z}_m=\{[0],[1],[2],\dots,[m-1]\}$, and I sort of understand that.. but I have no idea what ...
0
votes
1answer
79 views

Equivalence Relations using Quotients

I need help with this problem: Let $A$ resp. $B$ be a set, endowed with an equivalence relation $\sim_A$ resp. $\sim_B$. Defne a relation $\sim$ on $A \times B$ by setting $$(a_1, b_1) \sim (a_2, ...
1
vote
0answers
85 views

Function on equivalence relation on functions

Let $E$ be the set of all functions from a set $X$ into a set $Y$. Let $b \in X$ and let $R$ be the subset of $E \times E$ consisting of those pairs $(f,g)$ such that $f(b) = g(b)$. Prove that $R$ is ...
1
vote
1answer
117 views

Functions that are compatible with equivalence relations over its domain and codomain

Given a function $f : X \to Y$, an equivalence relation $\sim_X$ on $X$ and an equivalence relation $\sim_Y$ on $Y$, there is a notion of ``compatibility'' between $f$, $\sim_X$ and $\sim_Y$ if the ...