1
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0answers
29 views

A partial order with more properties than would be expected

Consider the relation: $$\langle x_1,x_2\rangle\prec\langle y_1,y_2\rangle\iff x_1,x_2,y_1,y_2\in\Bbb N\wedge x_1y_2<x_2y_1.$$ This is usually used for defining the (positive) fractions $\Bbb ...
1
vote
0answers
28 views

Prove/disprove questions on equivalence relations and ordered sets

If $R$ is an equivalence relation and a partial order over $A \neq \emptyset$ then every equivalence class contain at least one element. If $(A,\le)$ an ordered set, and $a\in A$ is a single ...
0
votes
0answers
18 views

Bijection between partial order and $<$

How to show the correspondence between a less than relation and partial orders ? here A less than relation $<$ on a set $S$ is a relation that satisfies If $a < b$ , then $a \neq b$. ...
1
vote
0answers
34 views

The null set as the underlying set of a relation structure

Can the null-set underly a relation structure? Can it underly a well-order? If so, would such a well-order have order-type $0$? (Can one even have an order-type of $0$?) My motivation for this ...
1
vote
2answers
336 views

equivalence relations and partial ordering

Let $A$ be a set with $6$ elements, $R$ be a relation on $A$ and $n = |\{(x, y) \in A \times A : xRy\}|$. (a) If $R$ is an equivalence relation on $A$, then what is the maximum value of $n$? (b) If ...
0
votes
2answers
342 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 ...