Explain the mathematical definition of a cryptosystem using words
The direct translation would be something like :


  For each key $e$ in the set of keys $\mathcal{K}$, there is a key $d$ also in the set of keys $\mathcal{K}$ such that the decryption function using $d$ as a key, called $D_d$, when applied to the encryption function using $e$ as a key called $E_e$ is the identity function, so that when both are applied in a succession, starting with $E_e$, to a plaintext $p$, one will recover the same plaintext $p$.


So, for each encryption key $e$, there is a decryption key $d$ (possibly the same key) which allows to decrypt the content encrypted using the key $e$.