Subsection 6.1.5 (00JJ): Total Relations

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

A relation $R : A \to | B$ is total if, for each $a \in A$ , we have $R (a) \neq Ø$ .

Let $R : A \to | B$ be a relation.

1. Characterisations. The following conditions are equivalent:
1. (a) The relation $R$ is total.
2. (b) We have $χ_{A} \subset R^{†} ⋄ R$ .

Item 1: Characterisations

We claim that Item (a) and Item (b) are indeed equivalent:

Item (a) $⟹$ Item (b): We have to show that, for each $(a, a^{'}) \in A$ , we have
$χ_{A} (a, a^{'}) ⪯_{{t, f}} [R^{†} ⋄ R] (a, a^{'}),$

i.e. that if $a = a^{'}$ , then there exists some $b \in B$ such that $a \sim_{R} b$ and $b \sim_{R^{†}} a^{'}$ (i.e. $a \sim_{R} b$ again), which follows from the totality of $R$ .
Item (b) $⟹$ Item (a): Given $a \in A$ , since $χ_{A} \subset R^{†} ⋄ R$ , we must have

${a} \subset [R^{†} ⋄ R] (a),$

implying that there must exist some $b \in B$ such that $a \sim_{R} b$ and $b \sim_{R^{†}} a$ (i.e. $a \sim_{R} b$ ) and thus $R (a) \neq Ø$ , as $b \in R (a)$ .

This finishes the proof.

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