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.