Why is $p(z) = \frac{e^z}{1 + e^z} \color{red}{\equiv} \frac{1}{1 + e^{-z}}$ and not $=$?

Question

Is it because of the division of the whole term by $e^z$? So, it would not be allowed to write $=$?

$\displaystyle \frac{1}{1 + 2^{-1}} = \frac{1}{1 + 0.5} = \frac{1}{1.5}$ has to be written with $=$, but it may not be written with $\equiv$. Correct?

In the second example, it is just a mathematical transformation without dividing the whole thing by something. In the first example, however, this seems to be different.

$\displaystyle p(z) = \frac{e^z}{1 + e^z} \equiv \frac{1}{1 + e^{-z}}$ shall represent a logistic function.

The context of everything is empirical research in economics.

Answer 1

The symbol “$=$” means a lot of different things. That includes, but is not limited to:

1. Equality/identity: Two things that are defined in different ways are in fact always equal. Example: $x^m \cdot x^n = x^{m+n}$.
2. Definition: You have some expression, and either it’s cumbersome to write it all the time or you want to be able to refer to it specifically without calling it “that expression, you know”. So, you want to give it a shorter name. Example: $e = \lim_{x\to \infty}\left(1+\frac1n\right)^n$.
3. Equation: You have two expressions that aren’t always equal, but you want to assume that they are equal and see what conclusions you can draw. Example: $x + 5 = 2x$, with the conclusion that $x$ must be equal to $5$.

Now, when you deal with expressions using letters (especially $x, y$ and $z$ for cultural reasons), point 1 and 3 may be confused. When you write $\frac{e^z}{1 + e^z} = \frac{1}{1 + e^{-z}}$, do you mean to say that the function of $z$ on the left-hand side is the same function as the one on the right? Or are you saying that you assume that $z$ is such that the value of the left side and the right side are equal, and you want to work out what consequences you get from it? (This interpretation is common when looking for intersections between two graphs.)

One way to solve this is by using more symbols. For that reason, some authors use $\equiv$ for situations like point 1 (at least when there is danger of confusion) and $=$ for 3. Similarily, you can sometimes see people use $:=$ for point 2.