Overloaded equal sign = and the word is

[Alain De Vos]

Problem,the equal symbol "=" in math is overloaded , as is the word "is" in language.
Is there a list of all the meanings of = in math and the word is.
For instance "is" can mean equal to , or element of a set.
Are there generally accepted alternatives to the "=" sign, so one can be more specific.
For instance := for equal per definition.

 

[disregardthat]

Why is it a problem that it's used extensively? As long as there is no ambiguity revolving around its use, I don't see how it could be a problem.

 

[Dr. Seafood]

Notation is simply about communicating ideas. So in an expression like 2 + 3 = 5, it's pretty clear what "=" denotes. When talking about isomorphism, congruence or some other type of "equality", :=, ≡, ⇔, ≈, etc. are used accordingly.

I find there's no need to be more specific. Mathematics is perfectly scrutinizing as it is (I don't mean that in a bad way). These are just symbols; a good math paper would be sure that there's no ambiguity in notation.

By the way, you wouldn't say "x is S" to mean x ∈ S, right? You would say "x is in S".

 

[SteveL27]

Problem,the equal symbol "=" in math is overloaded , as is the word "is" in language.
Is there a list of all the meanings of = in math and the word is.
For instance "is" can mean equal to , or element of a set.
Are there generally accepted alternatives to the "=" sign, so one can be more specific.
For instance := for equal per definition.

Actually, = is never overloaded in math.

It's an assignment operator in some programming languages; but every language that uses = for assignment uses == for equality, so there's confusion sometimes, but never overloading within a programming language or within math.

In typeset math they use $\equiv$ for mod equivalence, and $\cong$ for isomorphism. They never use = for anything but equality, as far as I know. Of course I could be wrong ... but I really don't think there are many alternate meanings or usage of = in math.

I think the main area where = is overloaded is in writing ASCII math, since there aren't any symbols for congruence, isomorphism, etc.

 

[CellCoree]

I can understand the confusion and potential for ambiguity when it comes to the equal sign "=" and the word "is" in mathematics and language. In mathematics, the equal sign is used to represent the relationship between two quantities that have the same value. However, it can also have other meanings depending on the context, such as indicating a definition or a logical statement.

Similarly, the word "is" in language can have multiple meanings, such as indicating existence or equivalence. In mathematics, it is commonly used to indicate an element of a set, but it can also be used in other contexts.

To answer your question, there is no definitive list of all the meanings of the equal sign in mathematics and the word "is" in language. However, there are generally accepted alternatives that can be used to be more specific and avoid confusion. For example, in mathematics, the symbol ":=" is often used to indicate an equal per definition, as you mentioned. Other symbols, such as "≡" for congruence or "≈" for approximation, can also be used to specify the type of equality being represented.

In language, using specific words or phrases can help clarify the meaning of "is" in a sentence. For instance, instead of saying "A is B," one could say "A is equivalent to B" or "A is an element of the set B."

Overall, it is important to be aware of the potential for confusion when using the equal sign and the word "is," and to use specific language and symbols to accurately convey the intended meaning.