1
vote
2answers
46 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
32 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
44 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
78 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
163 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
47 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
57 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
230 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
117 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
110 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
75 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
50 views

Functional relations : Trouble seeing transitivity

I've been 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 ...
4
votes
2answers
198 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
102 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
416 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
293 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
531 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
454 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
55 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
127 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
75 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
78 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
103 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 ...