1
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2answers
47 views

Geometric meaning of reflexive and symmetric relations

A relation $R$ on the set of real numbers can be thought of as a subset of the $xy$ plane. Moreover an equivalence relation on $S$ is determined by the subset $R$ of the set $S \times S$ consisting of ...
2
votes
1answer
25 views

partitioning a set using a non-equivalence relation

I understand that a set can be partitioned into equivalence classes by an equivalence relation (wikipedia article). However can we partition a set into a forest of trees by a relation that's simply ...
1
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0answers
27 views

Finding equivalence relations containing specific equivalences

"Find the number of equivalence relations on the set $\{1,2,3,\ldots,7\}$ such that: a) $1\sim2$ and $3\sim4$. b) $1\not\sim2$, $1\not\sim3$ and $3\not\sim2$." Solving this problem is equivalent to ...
1
vote
3answers
59 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
21 views

Equivalence Relations dealing with creating partitions.

Let R be a relation on a set A that is reflexive and transitive but not symmetric. Let R(x) = {y: xRy}. Does the set a = {R(x): x ∈ A} always form a partition of A? I really don't know where to ...
0
votes
1answer
48 views

Partitions on C = {i,-1,-i,1}

Let $ C = \{i, -1, -i, 1\}$ , where $ i^2 = -1 $. The relation $R$ on $C$ given by $xRy$ iff $xy = \pm 1$ is an equivalence relation on $C$. Give the partition of $C$ associated with $R$ I would ...
2
votes
2answers
79 views

Finest and Coarsest Equivalences

According to a theorem in the Set Theory book I am reading, we can understand that equivalence relations partitons a set $X$ into distinct equivalence classes, $[x]$. I get that, but one of the ...
1
vote
3answers
271 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.
2
votes
3answers
476 views

How can I find the partitions of this equivalence relation?

I have the following equivalence relation: $$\{(1,1),(1,4), (2,2), (3,3), (4,1), (4,4)\}$$ On the set: $ A = \{1,2,3,4\}$ How can I find it's partitions? This example will help me understand the ...
3
votes
3answers
1k views

Number of equivalence relations on a finite set

I need a hint for computing the number of ways in which all the equivalent classes on a set of $n$ elements can be realized. For example, if the set has 2 elements ${a,b}$, then there are 2 possible ...
3
votes
2answers
341 views

Equivalence relations on natural numbers

How many equivalence relations are there on $\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers? How can one compute it?
1
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2answers
930 views

Equivalence Relations and Partitions

My university "textbook" for discrete math is Schaum's Outline. In this outline he goes over Equivalence Relations and Partitions, and I got confused at a particular theorem. From the book: Theorem ...