Set of real numbers

All real numbers form the uncountable set ℝ. Among its subsets, relatively simple are the convex sets, each expressed as a range between two real numbers a and b where a ≤ b. There are actually four cases for the meaning of "between", depending on open or closed boundary:

[a, b]: {x | a ≤ x and x ≤ b }
(a, b): {x | a < x and x < b }
[a, b): {x | a ≤ x and x < b }
(a, b]: {x | a < x and x ≤ b }

Note that if a = b, of the four only [a, a] would be non-empty.

Task

Create a datatype that can represent any set of real numbers, for the definition of 'any' in the implementation notes below.
Provide methods for these common set opertions (x is a real number; A and B are sets):

x ∈ A: determine if x is an element of A

example: 1 is in [1, 2), while 2, 3, ... are not.

A ∪ B: union of A and B, i.e. {x | x ∈ A or x ∈ B}

example: [0, 2) ∪ (1, 3) = [0, 3); [0, 1) ∪ (2, 3] = well, [0, 1) ∪ (2, 3]

A ∩ B: intersection of A and B, i.e. {x | x ∈ A and x ∈ B}

example: [0, 2) ∩ (1, 3) = (1, 2); [0, 1) ∩ (2, 3] = empty set

A \ B: difference between A and B, also written as A − B, i.e. {x | x ∈ A and x ∉ B}

example: [0, 2) − (1, 3) = [0, 1]

Test your implementation by checking if numbers 0, 1, and 2 are in any of the following sets:

(0, 1] ∪ [0, 2)
[0, 2) ∩ (1, 2]
[0, 3) − (0, 1)
[0, 3) − [0, 1]

Implementation notes

'Any' real set means 'sets that can be expressed as the union of a finite number of convex real sets'. Cantor's set needs not apply.
Infinities should be handled gracefully; indeterminate numbers (NaN) can be ignored.
You can use your machine's native real number representation, which is probably IEEE floating point, and assume it's good enough (it usually is).

Optional work

Define A = {x | 0 < x < 10 and |sin(π x²)| > 1/2 }, B = {x | 0 < x < 10 and |sin(π x)| > 1/2}, calculate the length of the real axis covered by the set A - B.