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user12262
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In the definition of functions, the explicit declaration of domain and of codomain in relation to each other is nicely accomplished by the arrow notation; e.g. considering the real tangent function:

$$\text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \longrightarrow \mathbb R,$$

which, being more specificly a surjective function (i.e. with its image being the entire codomain) can more specificly be denoted as

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \twoheadrightarrow \mathbb R; \quad \quad \quad \quad \quad \quad \quad \quad \quad(\text{surj-non-inj})$$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \twoheadrightarrow \mathbb R; \quad \quad \quad \quad \quad(\text{surj-non-inj})$

or for bijections (surjective as well as injective, i.e. one-to-one function) specificly

$ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{bij})}_{(\text{branch})} : (0 ... \pi / 2) \longleftrightarrow \mathbb R_+. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\text{surj-inj})$$ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{bij})}_{(\text{branch})} : (0 ... \pi / 2) \longleftrightarrow \mathbb R_+. \quad \quad \quad \quad \quad \quad \quad \quad \quad (\text{surj-inj})$

However, of course we may also consider and purposefully declare functional relations (necessarily with explicitly specified domain and codomain) which are not surjective, e.g. the injective function:

$\quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{branch})} \quad \text{ again with domain } (0 ... \pi / 2) \quad \text{ but codomain } \mathbb R, \quad \quad \quad \quad \quad \quad (\text{inj-non-surj})$$\quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{branch})} \quad \text{ again with domain } (0 ... \pi / 2) \quad \text{ but codomain } \mathbb R, \quad \quad (\text{inj-non-surj})$

or the non-injective function:

$\quad \quad \quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{full})} \quad \text{ again with domain } \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \quad \text{ but codomain } \mathbb C. \quad \quad \quad (\text{non-inj-non-surj}).$$\text{Tan}^{(\text{non-surj})}_{(\text{full})} \text{ again with domain } \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \text{but codomain } \mathbb C. \quad (\text{non-inj-non-surj}).$

My question:

Are there specific distinct arrow symbols known to have been used for expressing e.g. the function declarations $(\text{inj-non-surj})$ and $(\text{non-inj-non-surj})$ explicitly and distinctly in arrow notation ?

In the definition of functions, the explicit declaration of domain and of codomain in relation to each other is nicely accomplished by the arrow notation; e.g. considering the real tangent function:

$$\text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \longrightarrow \mathbb R,$$

which, being more specificly a surjective function (i.e. with its image being the entire codomain) can more specificly be denoted as

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \twoheadrightarrow \mathbb R; \quad \quad \quad \quad \quad \quad \quad \quad \quad(\text{surj-non-inj})$

or for bijections (surjective as well as injective, i.e. one-to-one function) specificly

$ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{bij})}_{(\text{branch})} : (0 ... \pi / 2) \longleftrightarrow \mathbb R_+. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\text{surj-inj})$

However, of course we may also consider and purposefully declare functional relations (necessarily with explicitly specified domain and codomain) which are not surjective, e.g. the injective function:

$\quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{branch})} \quad \text{ again with domain } (0 ... \pi / 2) \quad \text{ but codomain } \mathbb R, \quad \quad \quad \quad \quad \quad (\text{inj-non-surj})$

or the non-injective function:

$\quad \quad \quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{full})} \quad \text{ again with domain } \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \quad \text{ but codomain } \mathbb C. \quad \quad \quad (\text{non-inj-non-surj}).$

My question:

Are there specific distinct arrow symbols known to have been used for expressing e.g. the function declarations $(\text{inj-non-surj})$ and $(\text{non-inj-non-surj})$ explicitly and distinctly in arrow notation ?

In the definition of functions, the explicit declaration of domain and of codomain in relation to each other is nicely accomplished by the arrow notation; e.g. considering the real tangent function:

$$\text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \longrightarrow \mathbb R,$$

which, being more specificly a surjective function (i.e. with its image being the entire codomain) can more specificly be denoted as

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \twoheadrightarrow \mathbb R; \quad \quad \quad \quad \quad(\text{surj-non-inj})$

or for bijections (surjective as well as injective, i.e. one-to-one function) specificly

$ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{bij})}_{(\text{branch})} : (0 ... \pi / 2) \longleftrightarrow \mathbb R_+. \quad \quad \quad \quad \quad \quad \quad \quad \quad (\text{surj-inj})$

However, of course we may also consider and purposefully declare functional relations (necessarily with explicitly specified domain and codomain) which are not surjective, e.g. the injective function:

$\quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{branch})} \quad \text{ again with domain } (0 ... \pi / 2) \quad \text{ but codomain } \mathbb R, \quad \quad (\text{inj-non-surj})$

or the non-injective function:

$\text{Tan}^{(\text{non-surj})}_{(\text{full})} \text{ again with domain } \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \text{but codomain } \mathbb C. \quad (\text{non-inj-non-surj}).$

My question:

Are there specific distinct arrow symbols known to have been used for expressing e.g. the function declarations $(\text{inj-non-surj})$ and $(\text{non-inj-non-surj})$ explicitly and distinctly in arrow notation ?

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user12262
user12262
  • 548
  • 2
  • 10

Arrow notation for distinguishing injective non-surjective from non-injective non-surjective functions

In the definition of functions, the explicit declaration of domain and of codomain in relation to each other is nicely accomplished by the arrow notation; e.g. considering the real tangent function:

$$\text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \longrightarrow \mathbb R,$$

which, being more specificly a surjective function (i.e. with its image being the entire codomain) can more specificly be denoted as

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}_{(\text{full})} : \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \twoheadrightarrow \mathbb R; \quad \quad \quad \quad \quad \quad \quad \quad \quad(\text{surj-non-inj})$

or for bijections (surjective as well as injective, i.e. one-to-one function) specificly

$ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{bij})}_{(\text{branch})} : (0 ... \pi / 2) \longleftrightarrow \mathbb R_+. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\text{surj-inj})$

However, of course we may also consider and purposefully declare functional relations (necessarily with explicitly specified domain and codomain) which are not surjective, e.g. the injective function:

$\quad \quad \quad \quad \quad \quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{branch})} \quad \text{ again with domain } (0 ... \pi / 2) \quad \text{ but codomain } \mathbb R, \quad \quad \quad \quad \quad \quad (\text{inj-non-surj})$

or the non-injective function:

$\quad \quad \quad \quad \quad \text{Tan}^{(\text{non-surj})}_{(\text{full})} \quad \text{ again with domain } \mathbb R \setminus \left\{ \frac{(2 \, k + 1) \, \pi}{2} \, | \, k \in \mathbb Z \right\} \quad \text{ but codomain } \mathbb C. \quad \quad \quad (\text{non-inj-non-surj}).$

My question:

Are there specific distinct arrow symbols known to have been used for expressing e.g. the function declarations $(\text{inj-non-surj})$ and $(\text{non-inj-non-surj})$ explicitly and distinctly in arrow notation ?