# Notation of a function that Maps two sets into a Matrix

Given two sets $P, V$ a function $f(t)$ takes any element that belongs to $P$ or $V$ e.g. $t \in P \cup V$ returns a matrix of $2$ columns and $K$ rows. What is the proper notation to express this function $f$ ? something like this ?

$$f: P\cup V \rightarrow ? X_{j,j} \;\forall j = 2, i=k$$

$K$ is finite, however not same for all $t \in P \cup V$ e.g. $K$ for $t_{1}$ and $K$ for $t_{2}$ may vary $\{t_{1}, t_{2}\} \in P\cup V$ like the following.

$$f(t_{1}) = \begin{bmatrix} v_{1}^{min} & v_{1}^{max} \\ v_{2}^{min} & v_{2}^{max} \\ v_{3}^{min} & v_{3}^{max} \end{bmatrix}, f(t_{2}) = \begin{bmatrix} v_{1}^{min} & v_{1}^{max} \\ v_{2}^{min} & v_{2}^{max} \end{bmatrix}, f(t_{3}) = \begin{bmatrix} v_{1}^{min} & v_{1}^{max} \end{bmatrix}$$

I can also express the matrix as a combination of two vectors $f_{min}$ and $f_{max}$ Then also How to express that number of rows of the matrix varies for $t_{i}$

And how to do the same with a Vector instead of matrix ? should I use $V$ instead of $M$ ?

-- Edit --

with the comments of @Jp McCarthy is it like $M_{2\times \vert f_{t_{i}} \vert}(\mathbb{Z}) \;\forall\; t_{i} \; \in (P \cup V)$ However this notation is not sometimes readable due to reduced font size

• $f:(P\cup V)\rightarrow M_{2\times K}(\mathbb{C})$ is one possible notation. – JP McCarthy Jun 9 '16 at 10:21
• What is C ? and is the arrow style correct ? – Neel Basu Jun 9 '16 at 10:25
• The arrow style is fine. $M_{2\times K}(\mathbb{C})$ is the set of matrices of size $2\times K$ with entries in the set of complex numbers. If your matrix entries are real numbers you could use $\mathbb{R}$ instead. What type of numbers are your matrix entries? – JP McCarthy Jun 9 '16 at 10:27
• Hm Real Numbers. – Neel Basu Jun 9 '16 at 10:30
• What if $K$ is derived ? e.g. K is not constant different elements of P or V will have different K ? something like $k = \vert f(x) \vert$ – Neel Basu Jun 9 '16 at 10:42