One of the reason for these sets is mathematics ℜ <> ℝ in a math text (and BTW the math symbols ℂℍℕℙℚℝ in the double strike set are "out of sequence" which can be a nasty surprise if you do naive incrementation.
And ℤ. The reason these double-struck symbols are in a weird place (U+2100-214f, separate from the rest in U+1d400-1d7ff) is because they all have commonly used special meanings in mathematics -- they're used to represent the sets of all numbers of various types. ℂ = complex numbers, ℍ = quaternions, ℕ = natural numbers, ℚ = rational numbers, ℝ = real numbers, ℤ = integers.