Given a set X, prove that there exists a free R-module F with a basis B for which there is a
bijection ϕ : B → X.
Consider the free module $\displaystyle R^{\oplus X}$ (i.e. $\displaystyle X$-fold coproduct). This has a natural basis of the form $\displaystyle \{e_x:x\in X\}$ where $\displaystyle e_x$ is the tuple with $\displaystyle 1$ in the $\displaystyle x^{\text{th}}$ coordinate and zero elsewhere. I think the rest should be obvious.