I am reading "A Course in Analysis vol.3" written by Kazuo Matsuzaka.
There is the following exercise in this book:
P.37 Exercise 13
Let $(X_i)_{i \in I}$ be a family of sets with $\lvert I \rvert = \aleph = \lvert \mathbb{R} \rvert$.
Assume $|X_i| = \aleph = \lvert \mathbb{R} \rvert$ for all $i \in I$.
Prove that $\lvert \displaystyle \bigcup_{i \in I} X_i \rvert = \aleph = \lvert \mathbb{R} \rvert$.
In the solution of this exercise, the author writes that "We can assume that $X_i \cap X_j = \emptyset$ for $i \neq j$ without loss of generality".
I cannot understand why we can assume that $X_i \cap X_j = \emptyset$ for $i \neq j$ without loss of generality.