The following properties of binary relations are fundamental to the concept of maps, which we will introduce later.

Definition: Total and Unique Binary Relations

Let $V, W$ be (not necessarily different) sets and $R\subseteq V\times W$ be a relation $R$ is called:

right-unique (also called functional, univalent or right-definite), if from $vRw_1$ and $vRw_2$ it follows that $w_1=w_2$ for the respective $v\in V$, $w_1,w_2\in W,$
left-unique (also called injective), if from $v_1Rw$ and $v_2Rw$ it follows that $v_1=v_2$ for the respective $v_1,v_2\in V$, $w\in W,$
unique if $R$ is right-unique and left-unique,
left-total, if for all $v\in V$ there is a $w\in W$ with $vRw$,
right-total, (also called surjective), if for all $w\in W$ there is a $v\in V$ with $vRw$,
total, if $R$ is left-total and right-total.
total, if $R$ is left-total and right-total.

Chapters: 1
Definitions: 2 3 4 5 6 7 8
Explanations: 9 10
Lemmas: 11 12
Motivations: 13
Proofs: 14 15 16 17 18 19 20 21 22
Propositions: 23

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Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001