Examples
- Z in R has no upper bound.
- Let the relation "≤" on {a, b, c, d} be given by a ≤ c, a ≤ d, b ≤ c, b ≤ d. The set {a, b} has upper bounds c and d, but no least upper bound.
- In Q, the set of numbers with their square less than 2 has upper bounds but no least upper bound.
- In R, the set of numbers less than 1 has a least upper bound, but no greatest element.
- In R, the set of numbers less than or equal to 1 has a greatest element.
- In R² with the product order, the set of (x, y) with 0 < x < 1 has no upper bound.
- In R² with the lexicographical order, this set has upper bounds, e.g. (1, 0). It has no least upper bound.
Read more about this topic: Greatest Element
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