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For me, it seems they all mean I can substitute one for the other in afterwards statements. Why do we invent so many terms? Why not just use one of them like "let $xxx=yyy$"? Is there any essential difference between them?

William
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    It's a matter of style. Sometimes you use all of these in a text. But should stay consistent, if you use := to define a formula, then you shouldn't use simply = anywhere unless you're denoting a consequential equality. – Daniel Donnelly Jul 18 '23 at 02:22
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    So: "Let $G$ be a group. We denote by $|G|$ the number of elements in the set $G$; We know that if $H$ is another finite group, such that $|H| = |G|$, and $f: G \to H$ is a group homomorphism that is surjective (alt. injective), then $G \approx H$. The converse is true; i.e. we have that $|G| = |H|$ and $f$ is a surjective (alt. injective) iff $G \approx H$. Let $g,h$ be two group homomorphisms whose respective domain & codomain are compatible, then we define $i := g \circ h$ [...]". Remember you want things to flow linguistically, and that comes more from English-math fluency. – Daniel Donnelly Jul 18 '23 at 02:30
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    See this post – Mauro ALLEGRANZA Jul 18 '23 at 09:41

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"Define", "denote", $=$, and $:=$, all can express the same idea: That one symbol is just a synonym (or abbreviation) for another. So when we say "Define $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$" nothing would change if we instead said "denote" or "let".

"Iff" can hold for things which are not equalities. For instance a polygon is a triangle if and only if it has three sides.

But also, the equal sign can not just define an equality, but also assert an equality. For instance, we could say "Assume that $2(x+1)=6$. Then $x+1=3$..." Neither of these equal signs are defining anything. Rather they are asserting (either by assumption, or inference) the equality of some two numbers. So not all of these expressions are always the same.

Similarly, in computer science we often use := to express that some piece of data is stored in some register. If your text is sufficiently oriented around computers, then it may express this by that symbol.

Addem
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