rank of the free group: $\mathrm{rank} F_X=|X|$
Solution 1:
Here is the argument written out in full so you can tell me which step you don't understand. Suppose $F_X$ is generated by a subset $Y$ with $|Y| < |X|$. This induces a surjection $F_Y \to F_X$. Abelianization gives a surjection $\text{Ab}(F_Y) \to \text{Ab}(F_X)$. Tensoring with $\mathbb{Q}$ gives a surjection from a vector space of dimension $|Y|$ to a vector space of dimension $|X|$; contradiction.