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:

1. 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 $v\in V$, $w_1,w_2\in W,$
2. left-unique (also called injective), if from $v_1Rw$ and $v_2Rw$ it follows that $v_1=v_2$ for the respective $v_1,v_2\in V$, $w\in W,$
3. unique if $R$ is right-unique and left-unique,
4. left-total, if for all $$v\in V$$ there is a $$w\in W$$ with $$vRw$$,
5. right-total, (also called surjective), if for all $$w\in W$$ there is a $$v\in V$$ with $$vRw$$,
6. total, if $R$ is left-total and right-total.
7. 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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References

Bibliography

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