Intervals as subsets of R

The number sets nest: N ⊂ Z ⊂ Q ⊂ R, where Q = {x : x = p/q, p, q ∈ Z, q ≠ 0}. The irrational numbers T = {x : x ∈ R and x ∉ Q} include √2, √5 and π; T ⊂ R and N ⊄ T.

For real a < b, the open interval (a, b) = {x : a < x < b} leaves out both end points, and the closed interval [a, b] = {x : a ≤ x ≤ b} keeps both.

The half-open intervals are [a, b) = {x : a ≤ x < b}, which keeps a and drops b, and (a, b] = {x : a < x ≤ b}, which drops a and keeps b.

Each of (a, b), [a, b], [a, b) and (a, b] has length b − a, and each contains infinitely many real numbers.

[0, ∞) is the set of non-negative reals, (−∞, 0) the set of negative reals and (−∞, ∞) the whole of R. The side at ∞ always takes a round bracket, because ∞ is not a real number that could be included.

Translating is routine: {x : x ∈ R, −2 < x ≤ 4} is the interval (−2, 4], and [1, 6) in set-builder form is {x : 1 ≤ x < 6}. On a number line an included end point is drawn as a filled dot and an excluded one as a hollow dot.