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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.

Nemgathos
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    What is the source / context of this $\equiv$? – Kim Jong Un Jan 15 '18 at 12:06
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    Usually $\equiv$ is used instead of $=$ to mean the equality is true for all values of some variable, here obviously $z$. That is, it's a functional identity, not an equation. You could as well write "$\dfrac{e^z}{1+e^z}=\dfrac{1}{1+e^{-z}},\forall z\in\Bbb C$". – Jean-Claude Arbaut Jan 15 '18 at 12:06
  • @KimJongUn It shall represent a logistic function $\displaystyle p(z) = \frac{e^z}{1 + e^z} \equiv \frac{1}{1 + e^{-z}}$. The context is empirical research in economics. – Nemgathos Jan 15 '18 at 12:10
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    In that case, I think it is a typo. Should be "$p(z)\equiv \ldots=\ldots$ ". I wouldn't overthink this particular one. – Kim Jong Un Jan 15 '18 at 12:12
  • @Jean-ClaudeArbaut What does $\mathbb{C}$ represent? – Nemgathos Jan 15 '18 at 12:13
  • @KimJongUn Can you explain why? – Nemgathos Jan 15 '18 at 12:14
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    Essentially, "$p(z)≡A=B$" says "Define $p(z)$ as $A$ which can also be written as $B$". – Kim Jong Un Jan 15 '18 at 12:15
  • @KimJongUn So, *acually* every function should be written as something like $f(x) \equiv x^2$ instead of $f(x) = x^2$? – Nemgathos Jan 15 '18 at 12:16
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    @Nemgathos Or, following my answer below, $f(x):=x^2$. That is something I have seen people write. – Arthur Jan 15 '18 at 12:18
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    The first time a notation is introduced ("$p(x)$" in your case), some authors use $\equiv$ to signify this first appearance and to tell the readers that this is how the notation is defined. Another example of this convention: $f(x)\equiv(x+1)^2=x^2+2x+1$. – Kim Jong Un Jan 15 '18 at 12:20
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    @Nemgathos $\Bbb C$ is the set of complex numbers, but you can take the set of real numbers instead if you wish. – Jean-Claude Arbaut Jan 15 '18 at 12:27
  • @Jean-ClaudeArbaut I cannot remember ever using $\mathbb{C}$, but now I know what it means. I think that our teachers have always used some other notation. But maybe I mix things up here. – Nemgathos Jan 15 '18 at 12:32
  • Multiply nominator and denominator with $e^{-z}$. Please note that the image of $e^{-z}$ is $\mathbb{C}\setminus\lbrace 0\rbrace$. The usage of $=$ or $\equiv$ depends on how you understand these both signs. – Fakemistake Jan 15 '18 at 12:35
  • By the way: Why does $z$ have to be a complex number? In my context, it is defined as $z_k = \beta_0 \sum_{j = 1}^{J} \beta_j x_{jk} + u_k$. $y_k = 1$ if $z_k > 0$ and $y_k = 0$ if $z_k < 0$. Is it because of $e$ itself? – Nemgathos Jan 15 '18 at 12:44

1 Answers1

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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.

Nemgathos
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Arthur
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  • very interesting... I never see this use of $\equiv$. I saw the symbol $\equiv$ for modular arithmetic and meta-logic with the meaning of same meaning for a string of symbols on both sides. – Masacroso Jan 15 '18 at 13:06