How to explicitly describe modifying variable assignments in FOL

Question

I have taken this question from the text "Language, Proof and Logic" by Barwise and Etchemendy (question 18.7, page 507).

The question reads as follows:

"(Modifying variable assignments.) Suppose $D = \{a,b,c,d\}$ and let $g$ be the variable assignment which is defined only on the variable $x$ and takes value $b$. Describe explicitly each of the following:"

1. $g[y/c]$
2. $g[x/c]$
3. $g[z/b]$
4. $g[x/b]$
5. $(g[x/c])[z/d]$
6. $(g[x/c])[x/d]$

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Progress so far:

From the text and this related question, I understand that the modifications assign:

$ \begin{array}{c|c|c|c} & g & x & y & z \\ \hline 1 & g[y/c] & b & c & \text{no assignment} \\ 2 & g[x/c] & c & \text{no assignment} & \text{no assignment} \\ 3 & g[z/b] & b & \text{no assignment} & b \\ 4 & g[x/b] & b & \text{no assignment} & \text{no assignment} \\ 5 & (g[x/c])[z/d] & c & \text{no assignment} & d \\ 6 & (g[x/c])[x/d] & d & \text{no assignment} & \text{no assignment} \\ \end{array} $

Where I am specifically uncertain is:

1) Is it correct to assume that for $\#4$ that this is the same as no modification?

2) What does the notation syntax in numbers $\#5 \ \& \ \# 6$ mean?

Edit: rows 5,6 of the table.

Answer 1

1) Well, it's still a modification, even though nothing changes. That is, the substitution of $b$ for $x$ still 'modifies' the original assignment of $b$ for $x$. But yes, effectively nothing changes

2) It's a modification of a modified assignment. That is, $g[x/c]$ is a modified assignment of $g$, but also an assignment in and of itself, and hence can be modified as well. So, for example, in $6$, we take the original $g$, which assignes $b$ to $x$, but then this gets modified to $g[x/c]$, and so now $c$ gets assigned to $x$, but then we modify that assignment in turn to $(g[x/c])[x/d]$, and so in the end $d$ ends up being assigned to $x$.