What Does It Mean When We Define a Variable in Mathematics?

[C0nfused]

Hi everybody,
All these letters that we use in our calculations are mostly called variables,parameters or name of functions, which are actually variables too. So how do we define variables? For example when we say that x is a real variable what exactly do we mean? Is it a letter that represents any element of the set, so in my example any real number? So if we have defined the set that the variable takes "values" from, then we can apply any operation of this set to specific elements of the set combined with the variable?

To make it more clear, if x is a real variable then we can do all the usual operations to expressions/equations/inequalities etc that contain the letter/variable x? So x is actually a real number and any operation of real numbers can apply to it (example 2*x+3*x=(2+3)*x=5*x=5x) ? This also applies to real functions? I mean f(x) is actually a number ,that can change so it's called a variable, but it's still a real number?

Similar to this, if x is a vector then we can do any operations defined for vectors with x ?

That's all,
Thanks

P.S: Something quite irrelevant: What exactly the equal sign "=" means? When we say that a=b this means that a is exactly the same thing as b?What's the difference with the other sign that i don't know its name, which is similar to "=" but with three lines?

 

[Hurkyl]

You're getting into formal logic. I think the best answer for you would be types...

A language has an assortment of type symbols, and a collection of variable symbols associated with each type.

So, for example, when we say "x is a real number", we mean that x is a variable symbol associated with the "real number" type.

The syntax of mathematics would then specify the types of various strings of symbols. For example, if:

f is of the "functions from R to R" type,
x is of the "real number" type,
Then,
the string of symbols f(x) is of the "real number" type.



One of the good points about set theory is that it's good at modelling this sort of language. So, each type corresponds to some set. E.G. there's a set of real numbers, and a set of functions from R to R. The string of symbols "f(x)" is interpreted as the evaluation of f at x.



There are a couple of ways to interpret the "=" symbol. One way is that if "a = b", then you can substitute b in for a in any formula. Another way is that "=" is simply another binary relation that satisfies some properties.