Let $V$ be a variety of algebra. Prove $F(X)$, the free algebra in $V$ with basis $X$ can be embedded in $F(Y)$ if $X \subset Y$.
I found this question on Chegg here while trying to study, but the answer is absolutely incorrect. They literally state that since $X \subset Y$, then $F(X) \subset F(Y)$ so clearly it can be embedded. I do think this is a really interesting question and something I should be able to show. Could someone please point me in the right direction? I know that $F(X)$ a free algebra means that $\forall A \in V, \forall \phi: X \rightarrow A,$ there exists a unique homomorphism $\bar{\phi}: F \rightarrow A$ that has the commutative diagram given by $X \rightarrow F$ (by $\iota), F \rightarrow A$ (by $\bar{\phi}$ and $X \rightarrow A ($ by $\phi)$. Am I just looking for an injective homomorphism or is there something else I need to show about the mapping?
