… my confusion can be summed up by asking for a breakdown in the difference of these 4 notations regarding [domain statements]:
Posting a question is
not a summation. :???: (By the way, you have asked about five forms, not four.)
I'm not sure what kind of "breakdown" you want. Except for the last one, they are all different forms of saying the same thing (after corrections, noted below).
If you remove the unneccessary parentheses, what remains is called Set-Builder Notation.
Here, your symbol I represents "the set of Integers".
We read this notation as, "The set of all x such that x is an Integer".
If you remove the unneccessary curly braces, what remains is a simple statement: "x is an Integer".
Here, symbol
ℤ represents "the set of Integers".
This form is the same as the previous one. The curly braces are unneccessary. Symbol
ℤ already represents a set.
Putting curly braces around it leads to: "the set of the set of Integers". That's redundant.
This form is called Interval Notation. It cannot be used to express the set of Integers.
What you have typed is read as, "all Real numbers".
The symbol for that set is
ℝ.
PS: In addition to symbol
ℤ for the set of Integers and symbol
ℝ for the set of Real numbers, the following double-stroke characters represent other common sets:
The set of Natural numbers:
ℕ
The set of Rational numbers
ℚ
The set of Complex numbers:
ℂ