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

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1answer
24 views

Proving a simple partially ordered set

I am losing my mind over this: (a) The relation $A=\{(1,1),(2,2),(3,3),(4,4),(3,2),(2,1),(3,1),(4,1)\}$ on the set $S=\{1,2,3,4\}.$ I'm having trouble figuring out if it's reflexive, symmetric, ...
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1answer
30 views

Equality of equivalence classes for congruence modulo 7

Let R be the relation of congruence modulo 7. Which of the following equivalence classes are equal? [35], [3], [−7], [12], [0], [−2], [17] I got 3) [35] = [-7] = [0], [3] = [17], [12] = [-2] ...
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2answers
33 views

Finding the equivalence class of this relation

I am having this relation: $$ A=\mathcal P(\mathbb {N} \diagdown 0) , $$ A~B :<=> min A = min B I haved already ...
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0answers
6 views

Prove that $cl_{eq}(R) = (cl_{sym}(R))^*$

Prove that $cl_{eq}(R) = (cl_{sym}(R))^*$ $cl_{eq}(R) = \bigcap\{S | $ S is an equivalence relation and $R \subseteq S\}$ is the equivalence closure of R. $R^* = \bigcap\{S | $ S is reflexive, ...
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1answer
23 views

Ordered Pair Proof

Below are two relations on $R.$ For each one, determine (with proof) whether or not it is an equivalence relation. If it is an equivalence relation, describe the partition it induces (i.e., describe ...
2
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0answers
40 views

Prove $[(n, k)] = 7^{*} \iff n - k = 7$

Prove $[(n, k)] = 7^{*} \iff n - k = 7$ given $7^{*} = [(8,1)] \in \mathbb{Z}$ and $k < n \in \mathbb{N}$ using any facts about addition in the Peano System. $k < n \in \mathbb{N}$ is ...
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1answer
17 views

Doman and range of a simple relation

Relation xRy if x≥y^2 (on real numbers), I'm assuming the domain is (o, infinity) and the range is all real numbers?
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1answer
44 views

Equivalence relation for set given as matrix

I need a hint. My task is to proof that ((a11, a12), (b11, b12)) ((a21, a22), (b21, b22)) ∈ R ⇔ a11 + a12 + a21 + a22 = b11 + b12 + b21 + b22 R is equivalnce relation. My problem is that I ...
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1answer
32 views

Define a relation ~ on ℝ² by (x,y)~(w,z) if x+y=w+z

So, it comes in two parts: a. Prove that ~ is an equivalence relation on ℝ². b. Give a geometric description of the partition of ℝ² formed by the equivalence classes. For a, I have to prove that ~ ...
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0answers
27 views

equivalnce relation for sets given as matrix [duplicate]

I need a hint. My task is to proof that ((a11, a12), (b11, b12)) ((a21, a22), (b21, b22)) ∈ R ⇔ a11 + a12 + a21 + a22 = b11 + b12 + b21 + b22 R is equivalnce relation. My problem is that I have ...
1
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1answer
26 views

Find the set of a given equivalence relation

What is the set $[4]$ but I haven't seen any examples in the text that describe how to approach a question such as this one. ...
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0answers
113 views

Equivalence relation over groups $a\asymp_sb :\rightarrow\exists n\in\Bbb Z:as^n=b$: terminology and decision problem

Let's define this relation over the elements of an infinite group $(G,\cdot,e)$ $$a\asymp_sb :\rightarrow\exists n\in\Bbb Z(as^n=b)$$ where $a^n$ is defined as follow 1)$a^0=e$ 2)$a^{n+1}=aa^n$ ...
0
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1answer
21 views

Non-empty intersection of equivalence classes.

I'm having troubles with the following exercise about equivalence classes on a defined set. Let $R$ be an equivalence relation on a set $A$. Given $a,b \in A$ prove the following statements are ...
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0answers
50 views

Showing that 2^(N-1) is equivalent to 1 mod N [closed]

My math professor gave us the following problem on a past exam and I didn't get it right then and I still don't know how do it: ...
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1answer
63 views

Being contractible in homotopy theory vs. homotopy type theory

I'm trying to clarify the notion of being contractible in homotopy theory vs. homotopy type theory. Is the following right? "In homotopy theory the real interval $[0,1]$, considered as a subset ...
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0answers
69 views

Is “to be married” a transitive relation?

If you define a relation on the set of people, given by $R=\{x,y : x\text{ is married with } y\}$. Is this relation transitive? I would say it depends: In the western culture: If $x$ is married with ...
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0answers
41 views

Well-defined functions using mod $p$ equivalence classes

Prove that if $m, n$ are elements in the set $\mathbb{Z}$ and $m \equiv n \pmod p$, then $m^2 \equiv n^2 \pmod p$. Also, is the function $f: [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = ...
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1answer
52 views

Equivalence classes of multiples of 3

I'm having a little trouble wrapping my head around the elements of the equivalence classes using the following definition: for m, n in N, define m ~ n if m^2 - n^2 is a multiple of 3. 1) List four ...
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2answers
41 views

Well-defined functions on equivalence classes

Could someone please explain how to develop a proof and the reasoning behind the following: My professor said that to determine if a function is well-defined, we check to see if equivalent elements ...
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1answer
29 views

Equivalence Relation and functions question

Could someone please confirm if I understand this correctly? Here is the problem: define ~ on Z by m ~ n in case m^2 ~ n^2. 1) What, if anything, is wrong with the following "definition" of a ...
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2answers
113 views

How to show that $(a \sim b \iff n | a−b)$ is an equivalence relation? [closed]

Let $n \in \mathbb{Z}$, $n > 0$ be a fixed positive integer. Define the relation $\sim$ on the set $\mathbb{Z}$ of integers by setting $$ \forall a, b \in\mathbb{Z}\ (a \sim b \iff n | a−b). $$ ...
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1answer
27 views

to find the smallest and largest number of equivalence relation in a set

Let $s$ be a set of $n$ elements. The number of ordered pairs in the largest and smallest equivalence relation on set $s$ are $n^2$ and $n$. I am able to understand the largest set of equivalence ...
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1answer
31 views

Equivalence classes, relation, partitions

Let $\sim$ be an equivalence relation on M, and $M/{\sim}$ to be the partition of M by $\sim$, and $\sim_{M/\sim}$ to be the equivalence relation by the partition. How do I show that the two ...
1
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1answer
74 views

Is a subobject classifier logically equivalent to set-inclusion?

Can one subsume the notion of set-inclusion $\subseteq$ and $\subset$ with the notion of a subobject classifier, expressed via an injective morphism $\hookrightarrow$? Specifically: Are the ...
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4answers
65 views

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$ To prove an equivalence relation my guess is to show that reflexivity, ...
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2answers
30 views

Equivalence Relations on Set of Ordered Pairs

Let $\mathbb{R}$ be the relation on $\mathbb{Z} \times \mathbb{Z}$, that is elements of this relation are pairs of pairs of integers, such that $((a,b),(c,d)) \in \mathbb{R}$ if and only if $a-d = ...
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2answers
18 views

Quick question: functions to spaces with equivalence relations

So I'm a little confused about sending functions from spaces without equivalence relations to a space with equivalence relations. For example, I'm trying to define a function $f : S^{n} \rightarrow ...
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1answer
33 views

Restriction of an equivalence relation on a subset.

If we have an equivalence relation defined on a set E and S its subset. Is the relation defined on S is also an equivalence one? Thank you for your answers.
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1answer
133 views

determining reflective, symmetric, transitive, anti-symmetric properties and describing equivalence classes

The question: determine if p is reflective, symmetric, transitive and/or anti-symmetric, if p is an equivalence relation, describe the equivalence classes A = Z , and $apb$ if and only if $5 | (2 a + ...
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1answer
76 views

Let A be a non-empty set, and p an equivalence relation on A . Let a , b be an element of A . Prove that [ a ] = [ b ] is equivalent to apb

the question: a) Let A be a non-empty set, and p an equivalence relation on A . Let a , b be an element of A . Prove that [ a ] = [ b ] is equivalent to $apb$ b) If p is both an equivalence relation ...
1
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1answer
87 views

$A = \mathbb{R}$ , and $a\mathrel{p} b$ if and only if $\sin a = \sin b$

My question is: For the relation $p$ described below, determine if $p$ is reflexive, symmetric, transitive, anti-symmetric. In each case, if $p$ is an equivalence relation, describe the equivalence ...
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3answers
36 views

Given $n \sim r \iff n \equiv r \pmod d$, prove $\sim$ is an equivalence relation. [duplicate]

It is given that n belongs to Z and d belongs to N. How do I prove that n=r mod d defines equivalence relation? I know I have to prove it is reflexive, symmetric and transitive. But how do I do that? ...
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2answers
47 views

Can a relation be a partial order and an equivalence at the same time?

Can a relation be a partial order AND an equivalence at the same time? For instance, if we have a set A = {1, 2, 3, 4, 5} and a relation R on A defined as R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}: ...
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2answers
66 views

Does R^2 has the same property as R?

If R is a relation on set A, define $R^2$ by $aR^2b$ if and only if there exists c with aRc and cRb. If R is reflexive/symmetric/transitive does $R^2$ have the same property ? I'm not sure how to do ...
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2answers
34 views

prove the equivalence of the following statements: 2x-1 is irrational; x/3 is irrational

I am stumped. I really have no idea how to solve this problem. Can someone please help me through this? THE TWO EQUATIONS ARE SEPERATE
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1answer
48 views

An equivalence relation iff G≈H, where G and H are groups [duplicate]

Problem : Let $S$ be the relation G~H iff G is isomorphic to H. Show reflexive, transitivity and symmetric. First show G is automorphism, which will imply G~G. So the identity mapping gives us ...
0
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1answer
21 views

Possible Equivalence Relation Question

Consider $\langle\Bbb{Z}_6, +_6\rangle$. Let $a\sim b$ if and only if $\{a,b\}$ generates $\langle\Bbb{Z}_6, +_6\rangle$. $a,b \in \Bbb{Z}_6$. Is $\sim$ an equivalence relation? I know an ...
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1answer
47 views

Find all equivalence classes

Let R by a relation defined on pairs $(m,n)$ of integers $m$ and natural numbers $n$ by $(i,j) R (k,l)$ if $il=jk$. Prove that this is an equivalence relation and give the equivalence cases. Show ...
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2answers
63 views

Identifying laws in a discrete math example

I'm studying for my upcoming discrete math test and I'm having trouble understanding some equivalences I found in a book on the subject. I guess I'm not really familiar with these rules and I would ...
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1answer
33 views

How can i do this algebra question? [closed]

The question is show that the relation $a\sim b$ defined by $a\equiv b \bmod 7$ is an equivalence relation on $\mathbb{Z}$. How many equivalence classes are there? Let us call them $[0]$, $[1]$, ..., ...
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1answer
59 views

Double check $G\sim H$ iff $G≈H$

Let $S$ be the collection of all groups. Define a relation on $S$ by $G \sim H$ iff $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes. Proof: Let ...
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1answer
38 views

Proving an equivalence relation(specifically transitivity)

I'm currently learning about equivalence relations. I understand that an equivalence relation is a relation that is reflexive, symmetric, and transitive. But I'm having trouble proving the transitive ...
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1answer
25 views

equivalence Relation problem with some conditions

If A be a set with $|A|=n$. if R be a equivalence Relation on A and $|R|=r$, why $r-n$ always be even ?
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2answers
52 views

Equivalence Class Question

On the set $N\times N$ define $(m,n)\simeq(k,l)$ if $m+l=n+k$. Draw a sketch of $N\times N$ that shows several equivalence classes. (hint: sketch points on graph paper). I'm not quite sure how to ...
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0answers
25 views

What is the intersection of thses equivalence relations?

Let $S$ be the following subset of the plane: $$ S \colon= \{ \ (x,y) \ | \ y=x+1, \ 0 < x < 2 \ \}.$$ Then how to describe the equivalence relation $T$ on the real line that is the ...
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2answers
52 views

Equivalence relations for $\mathbb{N} \times \mathbb{N}$ question

On the set $\mathbb{N} \times \mathbb{N}$ define $(m, n) \sim (k, l)$ if $m + l = n + k$. Show that $\sim$ is an equivalence relation on $\mathbb{N} \times \mathbb{N}$. Draw a sketch of $\mathbb{N} ...
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1answer
104 views

Absolute Value Equivalence relation inequality Question

I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me? For $a, b ...
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1answer
25 views

equiv. class if aRb means a+b is a+b even

let s be set of integers. and say that aRb=a+b only if a+b is even. i've already shown that this is indeed a equivalance relation, but how to show its equivalance classes?
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1answer
40 views

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R?

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R? I don't even know how to start. I know to be a equivalence relation R must be reflexive, symmetric and ...
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2answers
60 views

Prove that $\mathbb{Q} \times \mathbb{Q}$ is countable.

Knowing that $\mathbb{Q}$ is countable, I must prove that $\mathbb{Q} \times \mathbb{Q}$ is countable. Teacher's proof: For each $a \in \mathbb{Q}$, let $A_a = \{(a,q) : q \in \mathbb{Q}\}$ so that ...