2

$A=\{1,2\}$

$B=\{2,3\}$

List all injective relations $F: A \rightarrow B$

This is what I came up with:

$$F_1(1)=2$$ $$F_1(2)=3$$ $$F_2(1)=3$$ $$F_2(2)=2$$

Is that all? It seems to me that I'm missing something here. Can you also have an element in $A$ that doesn't translate to any element in $B$?

TheHolyJoker
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Belen
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    You listed all injective functions (which are also relations, but not all) – Hagen von Eitzen Dec 15 '19 at 06:51
  • So, you can have, for example, $F_3:A\rightarrow B$, such that $F_3(1)=2$, and $ \exists!F_3(2)$? – Belen Dec 15 '19 at 06:59
  • Also, I saw this answer: https://math.stackexchange.com/questions/1937802/how-many-injective-relations-exist-between-a-and-b, and it seems that if |A|=|B|, and |A|=2, so there should be 2!=2 injective relations $F:A\rightarrow B$ – Belen Dec 15 '19 at 07:01
  • So, I listed 2 of them, so that's all of them. How come Hagen von Eitzen says there are more? – Belen Dec 15 '19 at 07:36
  • How is an injective relation defined? Does e.g. ${(1,2),,(1,3)}$ satisfy that definition? – Berci Dec 15 '19 at 10:59
  • This I don't know. Maybe someone could clarify this. – Belen Dec 15 '19 at 18:47

1 Answers1

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If what we want is the total amount of relations then there's a total of 9, considering that for any element $a \in A$, $F(a)$ might not exists, and $F(a)=\{ 2,3\}$ is also valid.

If we are talking about functions $F: A\rightarrow B$, there are only two, mentioned in the original question. Since there is an added restriction, that $\forall a \in A, \exists! b \hookrightarrow F(a)=b$.

Edit: disclaimer

Okay, I think this is logical since our proffessor said that there should be 9 relations in total, so I think that's the only way to come up with them. I'm not marking this answer as correct, though, since I'm still learning on this topic. It'd be helpful if someone with more experience could verify this.

Belen
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