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
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)