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This is a lazy question, but very often textbooks use the "$\equiv$" (equivalent to) sign and the "$:=$" (defined as) sign in the same places from book to book. I suppose equivalence to a previously defined concept is also a form of definition. Any rules/guidelines as to when to use which?

Related to this query, suppose I wished to indicate that a particular variable had a particular property without defining a set and using the inclusion "$\in$" notation - so, for example, if $A$ is a circle, I might want to write $A\equiv\bigcirc$" where $\bigcirc$ is somehow shorthand for the property of roundness. I know it sounds convoluted, but I am happy to elaborate my context if someone is interested. In particular, this sort of shorthand works well where a generic set definition is not easy to write.

Thanks.

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    @Bill, thanks for your reply. The terms you use are unfamiliar to me. Maybe something simple I can read to understand, say, what a "binding mechanism" is? I think I understand what you mean though. So here is the context - I can't promise it won't be confusing. Like one of the commentators said, I'd like to point out that my object has a certain property (like greenness) which might be difficult to quantify but (say) readily understood. The property of greenness is not a set - all objects having it is. I would like to have shorthand for "A is/inherits Green" by something like $A\equiv G$.2011-06-23

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When I write papers in logic, I often use notation such as $ \phi \equiv (\exists x) \psi(x) $ to define a formula $\phi$. I use $\equiv$ because the "=" symbol that might appear inside a formula, and so I want to use a different symbol to say that $\phi$ is defined as the formula $(\exists x)\psi(x)$. In this sort of definition, $\equiv$ does not mean"logically equivalent", it means the same thing that $:=$ does: the left side is defined to be the right side. I use $\leftrightarrow$ or $\Leftrightarrow$ for logical equivalence within a formula, so the $\equiv$ symbol could not appear there either. I don't use the $:=$ notation.

I am not the only person who uses this set of conventions, but at the same time they may be limited to the area of logic I write in. I would be very surprised to see an analysis book use $\equiv$ in this way.

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I've also seen this a lot, and I came to the conclusion that “$\equiv$” was simply misused as a definition symbol.

The notation $A\equiv B$ states that—in a certain sense—$A$ is “as mighty (or large) as” $B$, while you really seem to mean something like $A\in B$, like “$A$ belongs to the (set of) round objects”. Perhaps you could have a notation like $A\in\{\mathrm{round}\}\cap\{\mathrm{green}\}$ or $A\in\bigcirc\cap\color{green}{\spadesuit}$.

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I can answer to your question only in the formal approach. In Coq I use “Definition” to introduce a name for a new notion and “<->” (logical biconditional, logical equivalence) to say that some already defined notions are equivalent. When to use what is pretty intuitive there.

Definition implies equivalence, e.g.:

Definition incompatible (a b:Prop) := ~(a/\b). Theorem incompatible0 : forall a b:Prop, incompatible a b <-> ~(a/\b). Proof. firstorder. Qed. 

As for your example about roundness, I do not see how to use logical biconditional here. I would rather denote roundness with an 1-ary predicate. E.g. “x is a circle” is formally written $\operatorname{circle}(x)$.