1
vote
2answers
24 views

What is the cardinality of $\left|C_s\right|$?

Let $$C_s = \left\{ f\in \mathbb{N}/S \to \mathbb{N} : \forall M\in \mathbb{N} / S. f(M)\in M \right\}$$ Where $S$ is an equivalence class. I need to prove $$\left|C_s\right| > \aleph_0 \implies ...
3
votes
1answer
20 views

Going from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨~q)

I am confused on how to go from (p ∧ ~q) ∨ (~p ∧ q) to (p ∨ q) ∧ (~p ∨ ~q). I know they are equal because I plugged them into a truth table and all of the rows have the same values. What would be some ...
0
votes
2answers
25 views

equivalence properties of $\equiv \pmod n$ proofs

Prove the identity: $$a \equiv b \pmod n \wedge a\equiv c \pmod n\implies b\equiv c \pmod n$$ I need to prove this property of $\equiv \pmod{n}$ along with a few others can someone link me to a ...
3
votes
4answers
47 views

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, ...
-2
votes
1answer
55 views

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 ...
0
votes
2answers
217 views

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 ...
0
votes
2answers
54 views

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 $a \equiv b \pmod 5$. ...
0
votes
1answer
64 views

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 ...
0
votes
1answer
45 views

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 ...
0
votes
1answer
78 views

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 ...
0
votes
1answer
39 views

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. ...
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 $f(f(x))=x$ ...
1
vote
1answer
46 views

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 $f,g \in F$ $f R g \Leftrightarrow |f(A)|=|g(A)|$ $f(A)=\{f(x): x\in A\}$ ...
0
votes
1answer
61 views

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 ...
1
vote
3answers
62 views

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: ...
0
votes
1answer
46 views

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 ...
0
votes
1answer
45 views

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 ...
1
vote
1answer
71 views

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 ...
1
vote
2answers
274 views

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 ...
1
vote
3answers
720 views

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 ...
0
votes
1answer
64 views

$R\subset S\times S$ 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 ...
2
votes
3answers
300 views

Prove that any partition induces a unique equivalence relation.

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.
3
votes
3answers
3k views

Prove that the intersection of two equivalence relations is an equivalence relation.

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows. Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R ...
5
votes
1answer
75 views

Prove $x^2 + 2y^2 \neq 805, x,y\in\mathbb{Z}$, using equivalence classes modulo 5

Prove $x^2 + 2y^2 \neq 805, x,y\in\mathbb{Z}$. We did this in class and, for the life of me, I cannot remember how to finish the problem. It starts out by taking all of the values to be $\mod5$. ...
0
votes
2answers
350 views

how to show equivalence relation and its classes

i am stumbling again in proving things in maths. the task is to prove that this statement $A \sim B : \Longleftrightarrow \sum_{a\in A} a = \sum_{b\in B} b $ is an Equivalence Relation on Power Set ...