Proof:≤ is reflexive

Claim (see proof):

For any set [math]A [/math], we have [math]\href{/cs2800/wiki/index.php/Equality_(cardinality)}{ |A| ≤ |A|} [/math]. In other words, the "smaller than" relation on sets is reflexive.

Proof: ≤ is reflexive

Choose an arbitrary set [math]A [/math]. Then the identity function [math]id_A [/math] is an injection. To see this, suppose [math]\href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id}(a) = \href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id}(b) [/math]. Plugging in the definition of [math]\href{/cs2800/wiki/index.php?title=Id&action=edit&redlink=1}{id} [/math], this means that [math]a = b [/math] as required.