A-Level Real Intervals

A-Level Real Intervals

In A-Level mathematics, intervals are subsets of the real line ℝ. We usually write them using bracket notation (for example [a, b] or (a, b]) and we can also describe them using set-builder notation.

Let a, b ∈ ℝ with a < b. The standard real intervals are listed below.

1. Finite intervals

• Closed interval
  [a, b] = { x ∈ ℝ ∣ a ≤ x ≤ b }
• Open interval
  (a, b) = { x ∈ ℝ ∣ a < x < b }
• Half-open (left closed, right open)
  [a, b) = { x ∈ ℝ ∣ a ≤ x < b }
• Half-open (left open, right closed)
  (a, b] = { x ∈ ℝ ∣ a < x ≤ b }

2. Intervals unbounded above

• Closed at a, unbounded above
  [a, ∞) = { x ∈ ℝ ∣ x ≥ a }
• Open at a, unbounded above
  (a, ∞) = { x ∈ ℝ ∣ x > a }

3. Intervals unbounded below

• Unbounded below, closed at b
  (−∞, b] = { x ∈ ℝ ∣ x ≤ b }
• Unbounded below, open at b
  (−∞, b) = { x ∈ ℝ ∣ x < b }

4. Entire real line

• All real numbers
  (−∞, ∞) = ℝ = { x ∈ ℝ }

5. Special cases

• Single point (degenerate interval)
  [a, a] = { a }
• Empty interval
  (a, a) = ∅