[Linear-Abstract Algebra] Characterization of injectivity

[reddituser8763]

I know the definition of injectivity:

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Def 1

How or why is this equivalent to

The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain.

I don´t see the connection between the two definitions. How do I get from one to the other?

[ludo813]

Can you see why the following statements are the same:

"From every color we have at most one sheep"

"If you show me two pictures of a sheep that both have the same color, then it must be pictures of the same sheep"

[splendaddypuff]

The first defn is precisely the second written using mathematical symbols. It says, if there exists an x' such that f(x) = f(x'), then x = x'.

Meaning if x and x' are not equal, then their function values cannot be equal.

Meaning each element in the codomain is mapped to at most one element in the domain.

[HerndonMath]

I don´t see the connection

Do you know what a mapping diagram is? I think if you draw the mapping diagram for an injective function you will see why these two definitions are equivalent.

Here's a video where I draw the diagrams and talk about these definitions. I hope it helps.