How do I notate the conditional combination of two sequences?

Question

Apologies for any incorrect usage of maths notation in my explanation. I am not a mathematician, I operate in the area of computer science. So some dumbed down explanation and pointing out which bits are confusing or which bits I get wrong would be very helpful.

So I have two sequences $E$ and $X$ and $|E| \le |X|$ (size of E is smaller than or equal to size of X). I want to be able to show that given a certain condition, the elements in $E$ replace certain elements in $X$.

So far I've tried to show that every element $e_j \in E$ has an associated index $i$ such that $f:E \to I, f(e_j) = i$. I then want to show that some function like $g(e_j)$ returns a true or false that determines if $e_j$ is going to replace $x_{f(e_j)}$ such that:

\begin{equation}x^{\prime}_{f(e_j)} = \begin{cases} e_j,& \text{if } g(e_j) = T\\ x_{f(e_j)} & \text{otherwise} \end{cases}\end{equation}

I then want to also show that this process happens for all $e_j \in E$.

So the main questions are as follows:

1. Does my definition of function $f$ make sense and do I need to demonstrate that it is injective (i.e. no two $e_j$ can have the same $i$)?
2. How would I show that function $g$ returns a true or false?
3. Is using a function as a subscript valid since it returns an index?
4. To show that the process happens for all $e_j \in E$ should I put $\forall e_j \in E$ in front of the large equation above? Or is this unnecessary?
5. Is there any more appropriate language, or notation I could use that would have made this more clear and concise?

Answer 1

My suggestions:

Avoid using capital letters to denote sequences such as $E$ and $X$, this is very non-usual, I read this as expectancy operator and random variable $X$. You can use as you did $e_j$ to be a sequence element, but and represent the sequence with a parenthesis coupled with a domain for the index such as $(e_j), \, j\leq j_{max}$.

Also, make the domain and image of your function more comprehensible by stating $I\subset \mathbb{N}$. But also try avoiding to introduce an $i$ symbol like this. I though it was the complex unit.

Mathematicians also don't usually think with boolean variables (that have TRUE=T or FALSE=F as their values). Speak in terms of 0 and 1, preferably. Think that statements are true or false in a logical sense.

Answering your questions:

1-I think I understood your point, but defining $f(e_j) = i$ makes me believe it is a constant function. If I understood correctly it is a mapping between $E$ and $I$.

2- Zero for false and one for true are an option, but this also would depend on how the function $g$ works.

3-Yes, it is perfectly valid. Maybe you could define an auxiliary sequence defined as $b_k = f(e_j)$.

4- You could, but it would also need to be true. For instance you didn't specify $f$ to be injective hence there could be $j\neq j'$ such that $f(e_j)-f(e_{j'})$, and a desfavoable matching condition for $g$.

5-Dont think there would be something much more concise, also for something to be "clear" is a rather subjective question. I can propose the definition below and you may draw ideias you like form it:

Let: $(a_n), n\leq n_{max}$ and $b_k, k\leq k_{max}$ with $n_{max}\leq k_{max}$. Let $f(a_n)=f_n \in \mathbb{N} $ and $g(a_n)\in {0,1}$ Let $d_m, m\leq k_{max}$, be constructed by: $$ d_m = \left \{\begin{array}{cc} a_{m}, && \text{if} \, f_n=m \, \text{and} \, g(a_n) = 1 \\ b_m, && \text{otherwise}\end{array} \right. $$