Proving a new defined logic operator, using NOT, OR, AND, IMPLIES gates Question:

Define a new logical operator $A*B$. We will call this new binary operator $*$ a ”NAND” operation. The truth table for NAND is as follows: $$\begin{array}{|cc|c|}A&B&A*B\\\hline F&F&T\\F&T&T\\T&F&T\\T&T&F\end{array}$$ Prove that you can create all other logical operators we have discussed (NOT, OR, AND, IMPLIES) using only NAND operations. Hint: Start with creating a NOT operation using NAND and work from there.

I know how to get all the logic operators using the first part of the truth table A B, but how are you supposed to get it from the new NAND operator, cause your not proving its equivalence or proving a statement, maybe I'm reading the question wrong, can someone please explain this to me, thanks

You are discovering and then proving four equivalences. I’ll get you started: prove that $\neg A$ is logically equivalent to $A*A$. Once you have that, it’s pretty easy to get $A\land B$ solely in terms of the $*$ operator. Then you can go for $\lor$, and finally for $\to$.
