Step By Step Calculus » 2.5 - Bounds and Cardinality of Sets

Synopsis
Bounds of Sets: A set of real numbers A A is bounded above if there is a real number p p such that no number in A A is larger than p: p:
\displaystyle \exists p \; \mbox {such that} \; \forall x \in A, \quad x \leq p \; .
\displaystyle \exists p \; \mbox {such that} \; \forall x \in A, \quad x \leq p \; .
p p is called an upper bound of the set. Obviously if p p is an upper bound for a set, so is any number larger than p p.
The least upper bound of a set of real numbers is the smallest real number that is greater than or equal to every number in the set.
The least upper bound axiom says: “If a non-empty set of real numbers has an upper bound, then it has a least upper bound.” We define similar terms regarding the lower bound of a set. This axiom holds for the real numbers. It does not hold for the rational numbers.
Cardinality: The cardinal number of a finite set is the number of members in the set. We can “count" infinite sets by assigning to each a cardinal number. Two sets have the same cardinal number if their members can be matched up in a one-to-one manner with nothing left over in either set. The sets that can be matched one-to-one to the natural numbers are called “countable" and assigned the cardinal number \aleph_{0} \aleph_{0}. The real numbers are not countable. They are assigned the cardinal number c c.