How are Letters Used in Mathematics?

[C0nfused]

Use of letters in mathematics:
1) we use letters in mathematics to represent any element of a set. So
this letter (for example "x") is any element of a specific set and we call
it a variable. We can substitute it with any specific element of the set
(for example in functions) and we can also apply any properties that the
elements of this set have to expressions that contain this variable (for
example if x is a real variable then 3*x+4*x=(3+4)*x=7*x ).

2) we use letters when we refer to specific elements of a set that we just
don`t know ,don`t care which are they or that we are looking for ( in that
case they are called "unknown" ) . Also when we generally want to refer to
any element of a set, considering however that it is constant in every
specific case ( to make it more clear: f(x)=3*x+b x can vary but b is
considered a constant-however for the various values of b different
functions occur).

As you see, until now there`s no question . I think all these are correct
but want you to check these and correct any mistakes. Something to add:
when we say that a(bc)=(ab)c with a,b,c real numbers , a,b,c represent any
real number right? Are they variables? Also when we say that "if f is
continuous in [a,b] and f(a)f(b)<0 then there`s at least one j with
a<j<b and f(j)=0 (with a<b)" we mean that a,b can be any real
number with a<b but of course they are constants, right? I mean we
cannot put any value in the place of a,b so they are actually specific
numbers, but we use letters to show that in any (specific and defined) set
[a,b] we can apply this theorem?
Are there any other cases of using letters in mathematics?

Thanks

 

[QuantumQuest]

when we say that a(bc)=(ab)c with a,b,c real numbers , a,b,c represent any
real number right? Are they variables?

Yes, they are variables. They can take any real value - as long as we talk about real numbers, and when they all three do, we talk about the same property with constants.

Also when we say that "if f is
continuous in [a,b] and f(a)f(b)<0 then there`s at least one j with
a<j<b and f(j)=0 (with a<b)" we mean that a,b can be any real
number with a<b but of course they are constants, right? I mean we
cannot put any value in the place of a,b so they are actually specific
numbers, but we use letters to show that in any (specific and defined) set
[a,b] we can apply this theorem?

This is The Location of Roots Theorem. Here we have two conditions that must be satisfied: $f$ continuous over the closed interval $[a,b]$ and $f(a)f(b) \lt 0$, so the theorem applies for intervals with endpoints such that these conditions are met.