There, you'll find distinctions in the way the equality symbol is used in logics: FOL (First Order Logic) with identity (where = is a primitive logic symbol), and this is what your author intends when discussing $=$ as a logical symbol, which is its most common usage in first-order-logic). As a logical symbol, its incorporation adds the "Axioms ...

2015年5月6日 · Equality $=$ is usually used for equality. $\equiv$ is occasionally used for "identically equal to," which is in a sense stronger than equality, by denoting that the thing on the left and the thing on the right are equal in a sense that they are identities of each other.

There's equality at the meta-level (where we're talking about structures, interpretations, formulas and so forth), and the equality symbol that can appear in a formula. Unfortunately it is common not to distinguish visually between the two kinds of equals signs and instead let the reader figure out from the context which of them makes sense.

The symbol ≅ is used for isomorphism of objects of a category, and in particular for isomorphism of categories (which are objects of CAT). The symbol ≃ is used for equivalence of categories. At least, this is the convention used in this book and by most category theorists, although it is far from universal in mathematics at large.

The most common one however is $ := $. The symbol $\equiv$ is usually used to denote a logical equivalence. The symbol $\stackrel{\mathrm{def}}=$ should just be exiled along with $\div$. Ultimately, the symbol you choose is a matter of personal preference. I personally use $:=$.

2011年3月18日 · I have seen variants of these used by people who predate widespread knowledge of computer programming. It would be interesting to know the earliest uses of a special symbol for this (and what symbols were chosen). An advantage of $:=$ is that it has a partner, $=:$, allowing it to distinguish which side is equal to the other by definition.

2023年6月25日 · Let's talk about the FOL with equality as it seems the one most logic books talk about. For FOL without equality, it seems easier to understand the meaning of "=" which is interpreted as a predicate. logic

Is there a symbol for potential equality? Essentially I'd like to condense: $$ (a = b) \lor (a \ne b) $$ so that I can express the phrase "a may or may not be equal to b". Apologies if my syntax is not entirely correct; I come from a computer science background.

2017年6月29日 · I suppose similarity is an equivalence relation. $\equiv$ is "equivalent to" and appears in regularly in modular arithmetic. $\approx$ is aproximately. $\cong$ is congruence or isomorphism. And of course $=$ is equality. $\endgroup$ –

and "What is the relation between these two expressions?" and I was hoping for an answer to the latter. In any case their answer might be the same "There is a solution ⇔ These expressions have an equality", but I was hoping for some new notations to avoid having to assume things to begin with. $\endgroup$ –