If $y \in Y$ and $g \in Y^X$, we often write $y+g$ as shorthand for the map $x \mapsto y+ g(x)$. Similarly if $f \in Y^X$ then $f+g = x \mapsto f(x)+g(x)$. However this presupposes that we can distinguish between an element of $Y$ and an element of $Y^X$. That is, we require these sets be disjoint. Are they?



Set Exponentiation: Is Y always disjoint from Y^X? 
As Joel David Hamkins points out, the assertion is false.


"The fact is that $Y$ may have a function from $X$ to (some other part of) $Y$ as an element. For example, consider $Y=\mathrm{HC}$, the set of all hereditarily countable sets, and let $X=\omega$; observe in this case that $Y^X \subset Y$, since any function from $ω \rightarrow HC$ is itself hereditarily countable. Similar examples abound."


