# Tagged Questions

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### Equivalence Relation definitions of Coset - looks like 1-step Subgroup Test? [Fraleigh p. 97 theorem 10.1]

p. 4 We are especially interested in the case where the set is a group, and the equivalence relation has something to do with a given subgroup. That is, we want to partition a group G into subsets, ...
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### Let ~ be an equivalence relation on a set S. Show that b is an element of cl(a) <=> cl(a) = cl(b) (Where all a,b are elements of S)

This was a question on my last equivalence relations quiz and I'm not yet comfortable with the whole "class" idea. I understand that I must show transitivity, reflexivity and symmetry however I'm not ...
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### Determine the number of equivalence relations on the set {1, 2, 3, 4}

Hi this was a question listed on my last proofs and conjectures midterm. It is similar to my previous post however this asks a different question which is throwing me off.. Do I simply list all ...
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### Determining whether relations are equivalence classes, and finding the equivalence classes

Determine if each of the following relations is an equivalence relation. If so, determine the equivalence classes. $S = \Bbb Z$, $a \sim b \iff a \equiv b \pmod 3$ or $a \equiv b \pmod 5$. ...
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### Let $H=\{2^m: m ∈ Z\}$ Where $m$ is any integer, and $a\sim b\Leftrightarrow a/b$ is an element of $H$.

Show that is an equivalence relation and describe the elements in the equivalence class $\operatorname{cl}(3)$. We're studying sets and equivalence in my mathematical proofs class. As this is a ...
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### Let H be a (non-empty) subset of integers. Suppose a-b <=> a~b ∈ H is an equivalence relation.

Show that $0 \in H$. Show that if $a \in H$ then $-a \in H$, and lastly, if $a$ and $b \in H$ then $a + b ∈ H$. Last bonus question on our last unit test on sets, equivalence relations and proofs. I ...
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### If $X = \{1, 2, 3, 4\}$ show there are just two equivalence relations on $X$ with $1\sim 2$ and $2 \sim3$

If $X = \{1, 2, 3, 4\}$ and $\sim$ is an equivalence relation on $X$, then if $1 \sim 2$ and $2 \sim 3$ show that there are just two possibilities for the relation $\sim$ and describe both ...
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### Proving that $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$

This assingment is preparation for exam. I need to prove with $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$ that $\sim$ is equivalence relatio. Can you tell me how to do this. ...
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### 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 $f(f(x))=x$ ...
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### how many elements does Ia have?

Let $A=\{1,2,3,4\}$. Let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by $f,g \in F$ $f R g \Leftrightarrow |f(A)|=|g(A)|$ $f(A)=\{f(x): x\in A\}$ ...
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### Equivalence relation of a group acting on a set

Let A be a set and G be any subgroup of S(A). G is a group of permutations of A. Assume that G is a finite group. If u∈A, the orbit of u is the set O(u)={g(u): g∈G}. Define a relation ~ on A by u~v ...
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### Partition of an equivalence relation

I am having a hard time with the following problem: In F(R), let f~g iff f(x)=g(x) for all x>c where c is some fixed real number. I proved that it was a equivalence relation by the following: ...
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### Properties of equivalence relations

Let $\sim_1$ and $\sim_2$ be distinct equivalence relations on $A$. Define $\sim_3$ by $a\sim_1 b$ and $a\sim_2 b$. Let $[x]_i$ denote the equivalence class of $x$ for $\sim_i$ ($i=1,2,3$). Prove ...
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### Proving that the relation $(x,y)S(x',y') \iff x - x' \in \mathbb{Z} \land y = y'$ is of equivalence.

The relation $S$ is of equivalence. I have to prove it. I managed to prove reflexibility and transitivity, but I'm having problems with symmetry. How can I prove it? The relation $S$ is defined ...
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### Determining equivalence classes of certain pairs for the relation $(a,b)R(c,d) \iff a^2 + 7b^2 = c^2 +7d^2$

This is an equivalence relations exercise. It has two parts. The first is about proving that the relation is of equivalence, which seems to be fine to me, but I'll put it there anyway. With the second ...
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### Equivalence Relations Proof dealing 3 dividing x + y

Consider the relation $S$ on the Natural Numbers defined by $\quad x\,S\,y\quad$ if $3$ divides $\quad x + y.\quad$ Prove $S$ is not an equivalence relation. I know an equivalence relation is one ...
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### Prove that this is an equivalence relation and give all the different equivalence classes [closed]

Let $R$ be a relation defined on real numbers by letting $a\mathrel R b$ iff $\cos (a) = \cos (-b)$. Prove that this is an equivalence relation and give all the different equivalence classes. Also ...
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### $R\subset S\times S$ for $S=\{1,2,3\}$: A Graph-Theoretic Approach

So I am given the relation $R=\{(1,1),(2,2),(3,3),(1,2),(1,3)\}$ and asked which of the properties reflexive, symmetric, or transitive are held in the relation, but what I am thinking is that this can ...
Given any partition $D$ of $A$, $\exists !$ equivalence relation on $A$ from which it is derived. Can someone please help me solve this problem? thanks.