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1.4/1.5 Real vocab
1.4 Real vocab
| Term | Definition |
|---|---|
| Set bounded above and upper bound | A subset E of ℝ is bounded above if there exists β ∈ ℝ such that x ≤ β for every x ∈ E. Such a β is called an upper bound. |
| Set bounded below and lower bound | A subset E of ℝ is bounded below if there exists γ ∈ ℝ such that x ≥ γ for every x ∈ E. Such a γ is called an upper bound. |
| Supremum/Least upper bound | Let E be a nonempty subset of ℝ that is bounded above. An element α ∈ ℝ is called the least upper bound or supremum of E if 1. α is an upper bound of E, and 2. if β ∈ ℝ and β is an upper bound then β ≥ α denoted: α = sup E |
| Infimum/Greatest lower bound | Let E be a nonempty subset of ℝ that is bounded below. An element α ∈ ℝ is called the greatest lower bound or infimum of E if 1. α is a lower bound of E, and 2. if γ ∈ ℝ and γ is a lower bound then γ ≤ α denoted: α = inf E |
| Supremum Property | Every nonempty set of ℝ that is bounded above has a supremum in ℝ |
| Infimum Property | Every nonempty set of ℝ that is bounded below has an infimum in ℝ |
| Open interval | For a, b ∈ ℝ, a ≤ b, the open interval (a, b) is defined as (a, b) = {x ∈ ℝ: a < x < b} |
| Closed interval | For a, b ∈ ℝ, a ≤ b, the closed interval [a, b] is defined as [a, b] = {x ∈ ℝ: a ≤ x ≤ b} |
| Half-open (half-closed) intervals | For a, b ∈ ℝ, a ≤ b, the closed interval [a, b] is defined as [a, b) = {x ∈ ℝ: a ≤ x < b}, (a, b] = {x ∈ ℝ: a < x ≤ b} |
| Archimedean Property | If x, y ∈ ℝ and x > 0, then there exists a positive integer n such that nx > y |
Created by:
Mollygsmithh
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