What is the explanation of the constant function formula?

Question

I know a little bit about constant function that is enough for me to think about what it is.But my book "Discrete mathematics and its application" by Kenneth Rosen (Indian adaptation)" contains a formula about constant function that is:

For each $k\ge0$,and each $a\ge0$,the constant function $C^k_a:\mathbb N^k\to\mathbb N$ is defined by the formula $$C^k_a (x)=a \ \text { for every element } x \in \mathbb N^k$$ In the case $k=0$, we may identify the function $C^k_a$ with the number $a$

Literally,I am not familiar with $C^k_a$ and $\mathbb N^k$.What they are?

And what will be the explanation of this formula?

Thanks.

Answer 1

$\mathbb{N}^k$ is meant to be the $k$-fold Cartesian product. It is the set of all $k$-tuples of natural numbers.

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$C^k_a$ is a composite symbol. The quote is introducing a variable $C$, but the problem is introducing lots of variables $C$. The variables $k$ and $a$ are decorations indicating which $C$ is intended in the formula.

This notation is usually thought of as a 'sequence' or an 'indexed family' of things.

However, it is not unreasonable to think of it as an alternate notation for a function. If you do so, you think of $C$ as the function of two arguments, and $C^k_a$ is the evaluation of the function at $k$ and $a$. The value $C^k_a$ is not itself a number, but instead yet another function.

People usually have trouble with function-valued-functions at first, so this alternate way of thinking about it might not actually be helpful. It's convenient when you're used to it, though.

Answer 2

$\mathbb N^k$ is the the cartesian product of the Naturals $k$ times, i.e. $\mathbb N \times \mathbb N \times \mathbb N \times \ldots \times \mathbb N$

A concrete example, if $k=3$, $\mathbb N^3 = \{ (a,b,c) \}$ where $a,b,c \in \mathbb N$

The $C_a^k(X)$ is only a function that set the number $a$ for each element of the previous Cartesian product. The $a$ denotes the constant point and the $k$ denotes the quantity of elements of the Cartesian product.