I want to see how to prove the functional completeness of NAND gate, but all the materials in the web I have reached just relies on the fact that the set $\{AND,NOT\}$ is complete and shows how to simulate those gates only using NAND gate. I want to see the proof or some list of references which contains the proof.
Thanks in advance.
As confirmed by the OP, I'll simply write down the proof that $\{\land, \lnot\}$ is complete.
There are four possible choices for a pair of truth values (00, 01, 10, 11). There are $2^4 = 16$ possible functions from $\{00, 01, 10, 11\}$ to $\{0, 1\}$. For simplicity, I will represent a function by a sequence of its values on $00, 01, 10, 11$ respectively. For example, $0001$ is the AND function. $0111$ is the OR function.
Here is one way to construct 8 out of 16 binary functions using only $\land$ and $\lnot$: $$ \begin{array}{rl} 0000: & (p, q) \mapsto p \land \lnot p \\ 0001: & (p, q) \mapsto p \land q \\ 0010: & (p, q) \mapsto p \land \lnot q \\ 0011: & (p, q) \mapsto p \\ 0100: & (p, q) \mapsto \lnot p \land q \\ 0101: & (p, q) \mapsto q \\ 0110: & (p, q) \mapsto \lnot (p \land q) \land \lnot(\lnot p \land \lnot q) \\ 0111: & (p, q) \mapsto \lnot (\lnot p \land \lnot q) \end{array} $$ Now observe that if we negate each of these functions, we obtain all the remaining 8 functions.