Examples and Properties
- If X = {a, b, c} and Y = {apples, oranges, peaches}, then | X | = | Y | because {(a, apples), (b, oranges), (c, peaches)} is a bijection between the sets X and Y. The cardinality of each of X and Y is 3.
- If | X | < | Y |, then there exists Z such that | X | = | Z | and Z ⊆ Y.
- If | X | ≤ | Y | and | Y | ≤ | X |, then | X | = | Y |. This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem.
- Sets with cardinality of the continuum
Read more about this topic: Cardinality
Famous quotes containing the words examples and/or properties:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)