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I think this is easier to understand the mapping of injective, surjective and bijective in terms of marraige proposal where men are from set A and women from set B but Is this analogy correct?

Injective (One-to-One) Functions:

If the marriage process between men from set A and women in set B is injective, this means:

  • Every man proposes to a distinct one and only one woman
  • Some women might not receive any proposals (i.e., remain unmarried), but no woman receives proposals from multiple men.

So, in terms of our marriage analogy for injectivity: each man proposes to one woman, and no woman has more than one suitor.

Surjective (Onto) Functions:

If the marriage process is surjective, it implies:

  • Every woman receives at least one proposal.
  • All men have a marraige proposal and it's possible for a woman to have multiple suitors.

For surjectivity: every woman gets at least one proposal

Bijective Functions:

For a marriage process to be bijective, every man must propose to a distinct woman such that every woman gets exactly one proposal, and every proposal is accepted. No man or woman is left without a partner, and there's no situation where a woman has more than one suitor or vice versa.

To sum it up using the marriage analogy:

  • Injective: Every man has a partner (definitely unique since they are not allowed to propose multiple women), but some women might be left without a partner.
  • Surjective: Every woman has a partner and they are allowed to propose multiple men so that no man remains without a proposal
  • Bijective: Every man and every woman have unique partners. Nobody is left out, and no overlaps in pairings.
munish
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  • This is a bit unclear. What's the function? At first I thought it was "consider the function from the set of Men to the set of Women given by Man $n$ proposes to Woman $F(n)$." But then you bring acceptance into it. So is that part of the function or not? – lulu Sep 28 '23 at 10:33
  • A critical part of the definition of a "function" is that it has to be defined on its entire domain. If your function is $F:A\to B$ then $F(a)$ has to exist, uniquely, for every $a\in A$. Of course, it's possible that you meant to restrict the domain to some subset (possibly empty) of $A$, but you never said that. – lulu Sep 28 '23 at 10:36
  • "Surjective: [...] but some men might be left without a partner" No, except in the perspective of lulu's comment, but this has nothing to do with surjectivity. – Anne Bauval Sep 28 '23 at 10:41
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    Post edit, the ambiguity still remains. At some points you speak of men proposing to women, at others you speak of people getting married. The two are not synonyms. I think the details of your example are getting in your way. If you don't want to drop it entirely, then just stick to proposals (but then you have to make it clear that each man must propose to exactly one woman, though one woman may, in principle, receive many proposals). But why not just use abstract sets? Examples aren't good if they add more confusion than they remove. – lulu Sep 28 '23 at 11:17

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What do you mean any man can stay single? Because if the domain of the function is men, then there cannot be single men since it is a function. In fact, there could be women with two men.

In neither case can a man have more than one suitor, as this would no longer be a function.

It can happen that a woman chooses two men, in this way it would still be a function and could still be surjective.

Leandro Santos
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