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)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“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)