Cardinality of the set of non isomorphic structures of fixed cardinality

Question

Let $L$, be a language and $\alpha$ be a cardinal; let $\Gamma:= \{\text{set of non isomorphic $L$ structures, having cardinality $\alpha$}\}$. Prove that $\operatorname{Card}(\Gamma)\leq 2^{\alpha\cup \operatorname{Card}(L)}$. My idea was to show that there is an injective map from $\Gamma$, to the set of functions from $\alpha\cup \operatorname{Card}(L)$ to $2$, but i can't find the way to define it.

Answer 1

HINT: An upper bound on $\operatorname{Card}(\Gamma)$ is the number of distinct $L$-structures with underlying set $\alpha$. Each such structure is defined by its interpretation, which is a function

$$\iota:L\to\alpha\cup\bigcup_{n\ge 1}\wp(\alpha^n)\;:$$

constant symbols are sent to elements of $\alpha$, unary predicate symbols to elements of $\wp(\alpha)$, and so on.