Cardinality of the star-product and cardinality of union

Question

Let $\mathbb{H} \subseteq [0,1]$ be the set of all finite non-empty subsets of the interval [0,1]. Define the function $\star : \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{H}$ by: $$X \star Y = \{x \vee y \mid x \in X, y \in Y\}$$ where $\vee$ is the supremum with respect to usual order $\leq$.

I want to verify if $card(X \star Y) \leq card(X) + card(Y)$.

This question came up when I was looking through a book on discrete mathematics, and it has proved intractable so far. Any help would be much appreciated!

Answer 1

$$X \star Y = \{x \vee y \mid x \in X, y \in Y\} \subset X \cup Y $$

so $card(X \star Y) \leq card(X\cup Y) \leq card(X)+card(Y)$.