Editing Cardinal number (section)

=== Cardinal addition ===
If ''X'' and ''Y'' are [[Disjoint sets|disjoint]], addition is given by the [[union (set theory)|union]] of ''X'' and ''Y''.  If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace ''X'' by ''X''×{0} and ''Y'' by ''Y''×{1}).
:<math>|X| + |Y| = | X \cup Y|.</math><ref name=":0">{{harvnb|Schindler|2014|loc=pg. 34}}</ref>

Zero is an additive identity ''κ'' + 0 = 0 + ''κ'' = ''κ''.

Addition is [[associative]] (''κ'' + ''μ'') + ''ν'' = ''κ'' + (''μ'' + ''ν'').

Addition is [[commutative]] ''κ'' + ''μ'' = ''μ'' + ''κ''.

Addition is non-decreasing in both arguments:
:<math>(\kappa \le \mu) \rightarrow ((\kappa + \nu \le \mu + \nu) \mbox{ and } (\nu + \kappa \le \nu + \mu)).</math>

Assuming the axiom of choice, addition of infinite cardinal numbers is easy.  If either ''κ'' or ''μ'' is infinite, then
:<math>\kappa + \mu = \max\{\kappa, \mu\}\,.</math>

==== Subtraction ====
Assuming the axiom of choice and, given an infinite cardinal ''σ'' and a cardinal ''μ'', there exists a cardinal ''κ'' such that ''μ'' + ''κ'' = ''σ'' if and only if ''μ'' ≤ ''σ''. It will be unique (and equal to ''σ'') if and only if ''μ'' < ''σ''.