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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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    If you will provide a bit more context I will be glad to give it a try. But first, do explain - how does that relate to set theory?2011-06-22
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    Nope, no set theory - will edit to fix. :)2011-06-22
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    It would help if you supplied further context. Usually $:=$ denotes some form of parameter binding (not only for naming variables but also functions and higher-order objects). Rarely in mathematics is there any rigorous presentation of the denotation of such binding mechanisms. For that see e.g. the theory of denotational semantics of programming languages. On the other hand $\equiv$ is a much more overloaded symbol. For example, it can denote an arbitrary equivalence relation. It's denotation is highly context dependent.2011-06-22
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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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