let $C(x)$ be the free vector space generated by $X$.Then if $Y$ is a second set, $C(X \cup Y) = C(X) \oplus C(Y)$

Question

In the book of linear algebra by Werner Greub, at page $63$, it is asked that,

Let $X$ be any set and let $C(X)= \{f \mid f: X \to \Gamma$ such that $f(x) \not = 0$ for finitely many $x\in X$ $ \}$ be the free vector space generated by $X$. Show that if $Y$ is a second set, then $C(X \cup Y) = C(X) \oplus C(Y)$

It is clear that if $X \cap Y = \varnothing$, then the equality holds, but if it is not the case, the sum $C(X) \oplus C(Y)$ cannot be direct, so what am I missing in here ?

Answer 1

So as mentioned in the comments, the original problem seems to be phrased a bit imprecisely. "If $Y$ is a second set" should be changed to "If $Y$ is a set, such that $X\cap Y = \varnothing$". Otherwise consider the singleton set(s) $X = Y = \{*\}$. Then $X\cup Y = X$ and we would arrive at $$k \cong C(X)\cong C(X\cup Y) \cong C(X)\oplus C(Y)\cong k\oplus k,$$ a contradiction.