True or false: The contrapositive statement of "$\text{not } A\implies \text{not } B$" is "$A\implies B$"? Question : State whether 'True' or 'False' and justify your answer giving reason.
The contrapositive statement of &quot;$\text{not } A\implies \text{not } B$&quot; is &quot;$A\implies B$&quot; where $A$ and $B$ are two statements.

I know that when $A\implies B$ is equivalent to it's Contrapositive namely &quot;$\text{not } B\implies \text{not } A$&quot; that is if $B$ doesn't hold then $A$ doesn't hold.
I asked this question because I have never been taught this by my teacher and text given in my book is just given in 3 - 4 lines which is totally insufficient. Kindly help at least a bit.
Even if know that the answer is true but I don't know how to justify it.
I tried to take this example :
$ x\neq -2 \implies x^2\neq4$ is equivalent to it's Contrapositive $x = -2\implies x^2 = 4$
Well, the contrapositive of 
$$\neg A\longrightarrow \neg B$$
is $$\neg(\neg B)\longrightarrow \neg(\neg A)$$
but I think we're allowed to know that $\neg \neg A \equiv A, \neg\neg B \equiv B$, hence
the contrapostive is
$$B\longrightarrow A$$
which is not the same as $A\longrightarrow B$. So the answer is false.

In general, the law of contraposition says that the $P\longrightarrow Q$ and its contrapositive 
$$\neg Q\longrightarrow \neg P$$
are logically equivalent.



Using the example given, the contrapostive of
$ x\neq -2 \implies x^2\neq4$ 
is 
$$x^2 = 4\implies x = -2.$$
You can think like this :

The contrapositive of ' not A => not B ' is ' not( not A => not B ) ' which simplify to ' A not(=>) B ' which is ' A &lt;- B '