NOR and NAND operator tautology I am having trouble with a problem in the book I studied about logic that use the NOR operator (also known as Peirce's arrow) and the NAND operator.
The discrete math book said this is tautology $A↓(A↓A) \equiv T$.
We know that $(A↓A) \equiv \mathord\sim A$ where $ ↓$ is the NOR symbol.
so  $A↓(A↓A) \equiv A↓ \mathord\sim A \equiv \mathord\sim(A \lor \mathord\sim A) \equiv F$
but the book said  that since we know  $(A↓A) \equiv \mathord\sim A \equiv (A|A) $ where | is NAND symbol
$A↓(A↓A) \equiv A| \mathord\sim A \equiv \mathord\sim (A \land \mathord\sim A) \equiv\mathord\sim F \equiv T$ so it is a tautology.
So is it a tautology or not?
 A: A statement is a tautology if it is true under every possible interpretations of the literals that form it.  Here is the truth table for $\downarrow$.

So if $A$ is true, then $$A\downarrow(A\downarrow A) \equiv T\downarrow(T\downarrow T)\equiv T\downarrow F\equiv F$$ and if $A$ is false then $$A\downarrow(A\downarrow A) \equiv F\downarrow(F\downarrow F)\equiv F\downarrow T\equiv F$$
So, not only is $A\downarrow(A\downarrow A)$ not a tautology, it is a contradiction.
A: There seem to be many parts to what you're asking so I'll just break down your question:
"$A↓(A↓A) \equiv T$" 
is wrong. 
Instead it should be 
$A↓(A↓A) \equiv F$
which is an always-false statement. This is the opposite of a tautology and is known as a contradiction.
"we know that $(A↓A) \equiv \sim A$ where $ ↓$ is NOR symbol." 
Correct. 
"so  $A↓(A↓A) \equiv A↓ \sim A \equiv \sim(A \lor \sim A) \equiv F$" 
All correct. 
"but the book said  that since we know  $(A↓A) \equiv \sim A \equiv (A|A) $ where | is NAND symbol" 
This is correct and holds when both operands are identical.
"$A↓(A↓A) \equiv A| \sim A...$"
Wrong. It is unclear how you made that leap. 
"$...\equiv \sim (A \land \sim A) \equiv\sim F \equiv T$"
The rest of this chain of reasoning (after the wrong step) is correct but obviously the conclusion is wrong because of that earlier mistake.
I hope that clarifies things.
