# Tagged Questions

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### How many relations exist in the set of A

When A = {1,b,ø}, how many reflexive relations exist on the set? I have said that AxA={(1,1), (b,b), (ø,ø), (1,b), (1,ø), (b,1), (b,ø), (ø,1), (ø,b)} Would I be right in saying that there are only 3 ...
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### Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
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### Relational algebraic structures

Recently I came across the notion of relational $\beta$-algebra, defined as a set $S$ and a binary relation $\xi:\beta S-S$, where $\beta S$ denotes the set of ultrafilters on $S$ (and $\beta$ is the ...
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### The closures of a binary relation

I have the following dilemma concerning the equivalence closure of a binary relation. Let $X$ be a nonempty set and $R\subseteq X\times X$ (i.e., a binary relation over $X$). Consider the following ...
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### How do you call this feature and property of relation?

If an operator can be defined for $k$ operands, $k \in \mathbb N$, how do you call this feature of the operator? For example, "+" is such an operator. Similarly, for a relation $R$ on a set $X$, $R$ ...
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### Finite poset maximum and minimum element

Let $(P,\le)$ be a finite poset. An element $z \in P$ is an upper bound for $x,y \in P$ if $x \le z$ and $y \le z$. How do I prove that if every two elements in $P$ have an upper bound then ...
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### Let $S=\{a,b\}$. Of all the relations on $S$ which are symmetric? Reflexive? Transitive?

The relations are as follows: 1.) $\{(a,a)\}$ 2.) $\{(a,b)\}$ 3.) $\{(b,a)\}$ 4.) $\{(b,b)\}$ 5.) $\{(a,a),(a,b)\}$ 6.) $\{(a,a),(b,a)\}$ 7.) $\{(a,a),(b,b)\}$ 8.) $\{(a,b),(b,a)\}$ 9.) ...
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### Show that ≡ is an equivalence relation, Show that ⊕ is well-defined, and Show that ⊕ is a commutative and associative operation.

Let $(a,b),(x,y) \in\Bbb R\times\Bbb R$ and define $(a,b) \equiv (x,y)$ iff $a+b = x+y$. a. Show that $\equiv$ is an equivalence relation. Define the operation $\oplus$ on the equivalence classes as ...
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### Smallest Congruence Relation generated by a set

$\newcommand{\cl}{\operatorname{cl}}$ Let $R \subset S \times S$ be a binary relation, the smallest i) reflexive relation containing it is $$\cl_\mathrm{ref} = R \cup \{ (x,x) : x \in S \}$$ ii) ...
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### An analogy between subgroups and equivalence relations.

I have noticed a certain analogy between subgroups of a group $G$ and equivalence relations on a set $X$. I would like to know if there's an explanation for this analogy or a common generalization of ...
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### Can we extend the definition of a homomorphism to binary relations?

This is going to be quite a long post. The actual questions will be at the end of it in section "Questions." INTRODUCTION After receiving an answer to this question about extending the definition of ...
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### The fibres of a map form a partition of the domain.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
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### Two questions about equivalence relations

Question 1: Let $x,y \in S$ such that $x\sim y$ if $x^2 =y^2\pmod6$. Show that $\sim$ is an equivalence relation. This is what I tried: Reflexive: $x^2\pmod6 = x^2$ implying $x\sim x$ ...
This question is problem 11(a) in chapter 1 in 'Topics in Algebra' by I.N. Herstein. These are the properties of equivalence relation given in this book. Prop 1 $a \sim a$ Prop 2 $a \sim b$ ...