Between any two distinct real numbers a and b, where a
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Since they are distinct, we can assume without loss of generality that \(a < b\). Show more…
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Prerequisites
The Real Number Line and Order
Suppose a and b are real numbers with a < b. Recall the notion of intervals of real numbers: Interval Notation | Set Notation | Name (a, b) | {x ∈ ℑ | a < x < b} | Open interval [a, b] | {x ∈ ℑ | a ≤ x ≤ b} | Closed Interval [a, b) | {x ∈ ℑ | a ≤ x < b} | Half-open or half-closed interval (a, b] | {x ∈ ℑ | a < x ≤ b} | Half-open or half-closed interval (-∞, a) | {x ∈ ℑ | x < a} | Open ray (a, ∞) | {x ∈ ℑ | x > a} | Open ray (-∞, a] | {x ∈ ℑ | x ≤ a} | Closed ray [a, ∞) | {x ∈ ℑ | x ≥ a} | Closed ray (a) Is (a, b) a subset, proper subset, or neither, of (a, b]? (b) Is (a, b) a subset, proper subset, or neither, of (a, ∞)? (c) Use interval notation to describe the set [-3,7] ∩ (5,9). (d) Use interval notation to describe the set [-3,7] ∪ (5,9). (e) Write the set {x ∈ ℑ | |x| < 0.01} using interval notation. (f) Write the set {x ∈ ℑ | |x| > 100} as the union of two intervals.
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