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In other words, the surjection says: for any y in the codomain there should exist x in the domain. Now, do I need for every x in the domain to have an y in the codomain for surjectivity?

LearningMath
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    Domain = place where the function is defined. So, injective, surjective, or not, a function is defined for every point in its domain. – Matt Samuel May 25 '15 at 23:08
  • What about for a relation? I was under the impression that the difference between functions and relations are that functions have a single result value in the codomain for each domain value.
    And that relations had to be defined for all elements in the domain, similar to functions. I'm getting inconsistent and conflicting results when I google this.     – nickalh Nov 30 '23 at 09:40

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You need every $x$ in the domain to have a $y$ in the codomain because it's a function, and that's the definition of a function. It had nothing to do with being injective or surjective.

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    Here's a puzzler: what if the domain is empty? Look carefully at the definition of function and say if one can be defined on the empty set $\phi$. What happens if the codomain is empty? – Matematleta May 25 '15 at 23:10
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    @Chilango Not really a puzzler. Every function is defined on every point in the empty set, because there are no such points. – Matt Samuel May 25 '15 at 23:12
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    Right.But I was directing my post to the questioner who seems to be just starting out. You spoiled the fun. – Matematleta May 25 '15 at 23:15
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A function's surjectivity has got nothing to do with its domain and everything to do with its codomain.

If every element in its codomain Y is an image of one or more elements from its domain X, then the function is surjective.

Mathematically,

$\forall y \in Y, \exists x \in X \ \text{such that}\ y=f(x)$

Somesh Thakur
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Consider these 4 properties of a mathematical relation:

  1. Is every element in the Domain related to at least one element in the Co-Domain? (The "Defined for all Domain" property)
  2. For the elements in the Domain that are related to an element in the Co-Domain, are they related to only one element? (Well Defined)
  3. For the elements in the Co-Domain that are related to, is there only one element in the Domain that relates to them? (Injective)
  4. Do all of the elements in the Co-Domain have elements in the Domain related to them? (Surjective)

These are actually the same properties, but with the "arrows" pointing the opposite directions.

The if the entire Domain doesn't participate in the relation, then it isn't "Defined", and not a function. If the entire Co-Domain isn't involved, it isn't Surjective. If it isn't uniquely defined on the Domain, it isn't Well-Defined. If an element in the Co-Domain has many elements mapped to it, it isn't Injective.

If the sets have equal cardinality, then not being defined on the whole Domain means it can't be surjective, and if it isn't well-defined, then it can't be injective.

Soooo....

If the inverse function isn't defined for all elements, then it because the original function wasn't surjuctive, and if the inverse function isn't well defined, it is because the original function wasn't injective.

To be a Bijection, a relation must have all 4 properties.

Cool, huh?

Don Brandon
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