Proof:There is a unique function with an empty domain

From CS2800 wiki
For any set [math]A [/math], there is a unique function [math]f : \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} \href{/cs2800/wiki/index.php/%5Cto}{\to} A [/math].

This is an exercise in vacuous truth. In our logic, any statement of the form "for all [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} [/math], P" is automatically true (proof: choose an arbitrary [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} [/math], this is a contradiction, so we're done). We say that the statement is "vacuously true": it's technically true, but it doesn't say anything.

Using this, we can prove the claim:

Let [math]f : \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} \href{/cs2800/wiki/index.php/%5Cto}{\to} A [/math] be given by [math]f(x) := hamburger [/math]. To see that this is a function, we must check that for all [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} [/math], [math]hamburger \href{/cs2800/wiki/index.php/%5Cin}{\in} A [/math]. This statement is vacuously true. To see that [math]f [/math] is unique, suppose [math]f, g : \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} \href{/cs2800/wiki/index.php/%5Cto}{\to} A [/math]. We want to show that for all [math]x \href{/cs2800/wiki/index.php/%5Cin}{\in} \href{/cs2800/wiki/index.php?title=%5Cemptyset&action=edit&redlink=1}{\emptyset} [/math], [math]f(x) = g(x) [/math]. This is vacuously true.