# Tagged Questions

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### 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 ...
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### Partition induced by a Relation

Here's the problem: Let $A=\{1,2,3,4,5,6,7,8,9\}$. Define a relation $R$ on set $A$ by $xRy$ if and only if $2\mid(x+y)$ Assuming that $R$ is an equivalence relation, determine the partition of set ...
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### proving antisymmetry of partition refinement

Suppose $P$ is the set of all partitions of some set $S$. $R$ is a binary relation on $P$, the refinement relation, defined as $(\Pi_1,\Pi_2) \in R$ if and only if for every $S_1 \in \Pi_1$, there ...
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### Is this an equivalence relation?

I think the wording is throwing me off, and I also haven't done math in 4 months so basically my mind is scrambled eggs. Let $\sim$ be a relation on $\Bbb Z$ defined by letting $m \sim n$ if ...
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### 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 ...
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### General questions about equivalence classes and partitions

1) If a set is partitioned into non-overlapping, non-empty subsets, then those subsets are equivalence classes. If each element in a set is unique, how can a set be partitioned into subsets with ...
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### Proof of a Proposition on Partitions and Equivalence Classes

I stumbled upon a seemingly rudimentary proposition that I am having trouble writing out a proof for. The proposition goes something like, Proposition: If $\{A_i|i\in I\}$ is a partition of ...
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### Proving a relation between 2 sets as antisymmetric

Let $U = \{1,...,n\}$ And let $A$ and $B$ be partitions of the set $U$ such that: $\bigcup A = \bigcup B = U$ and $|A|=s, |B|=t$ Let's define a relation between the sets $A$ and $B$ as follows: \$B ...