Explain the mathematical definition of a cryptosystem using words So when googling "cryptosystem" I found a Wikipedia page which contained a mathematical definition of a cryptosystem. Find this definition below.

Could someone tell me what that what the last line means, in words instead of using mathematical signs? Maybe it will help my understanding of it.


  Mathematically, a cryptosystem or encryption scheme can be defined as a tuple $(\mathcal{P}, \mathcal{C},\mathcal{K},\mathcal{E},\mathcal{D})$ with the following properties.
  
  
  $\mathcal{P}$ is a set called the "plaintext space". Its elements are called plaintexts.
  $\mathcal{C}$ is a set called the "ciphertext space". Its elements are called ciphertexts.
  $\mathcal{K}$ is a set called the "key space". Its elements are called keys.
  $\mathcal{E}= \left\lbrace E_k:k\in \mathcal{K}\right\rbrace$ is a set of functions $E_k:\mathcal{P}\rightarrow \mathcal{C}$. Its elements are called "encryption functions".
  $\mathcal{D} = \left\lbrace D_k:k\in \mathcal{K}\right\rbrace$ is a set of functions $D_k:\mathcal{C}\rightarrow\mathcal{P}$. Its elements are called "decryption functions".
  
  
  For each $e\in \mathcal{K}$, there is $d\in \mathcal{K}$ such that $D_d(E_e(p))=p$ for all $p \in\mathcal{P}$.
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 $e$.