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I have the following definition:

Let $f$ be a function. We say that $f$ is injective if $(a, y)\in f$ and $(b,y)\in f$ (i.e., $f(a) = f(b)$) then $a = b$.

I understand the last sentence but I cannot establish the relationship between the definition and injective function.

Ѕᴀᴀᴅ
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Matheus
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  • (3) IS the definition of injectivity. So, if $f$ is a function such that (3) holds (for all $a,b$ in the domain) then we say $f$ is injective. – lanskey Mar 22 '18 at 00:53
  • Injectivity is also referred to as being "one-to-one" whereas a non-injective function is referred to as being "many-to-one". This might be more intuitive since it emphasizes that when the function is not injective, two values $a\neq b$ in the domain of $f$ can have the same function value, i.e., $f(a)=f(b)$ but when $f$ is injective, for every $y$ in the codomain of $f$, there is at most one $x$ in the domain of $f$ such that $f(x)=y$. Replacing "at most" by "at least" gives you the definition of surjectivity. Bijectivity holds when there exists precisely one such $x$. – Prasun Biswas Mar 22 '18 at 01:09

2 Answers2

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If $$ a=b \implies f(a)=f(b) $$

Then $f$ is a function.

If $$ f(a)=f(b) \implies a=b $$

then $f$ is injective.

For example $$f=\{ (1,2), (1,3)\}$$ is not a function.

$$f=\{ (1,2), (3,2)\}$$ is a function but it is not injective.

$$f=\{ (1,2), (2,3)\}$$ is a function. It is also injective.

Mohammad Riazi-Kermani
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This definition says that no two distinct points are mapped to the same value. You can rewrite this in terms of the contrapositive statement, saying that $f$ is injective iff $a\neq b \implies f(a) \neq f(b)$ for all $a,b$ in the domain.

terrygarcia
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