Consider $\alpha,\beta,\gamma$ as logical variables. Identify which of the following represents correct logical equivalence:

(A) $(\alpha\land(\beta\lor\gamma))\equiv((\alpha\land\beta)\lor(\alpha\land\gamma))$
(B) $(\alpha\lor\beta)\equiv\neg\alpha\lor\beta$
(C) $(\alpha\Rightarrow\beta)\equiv(\neg\beta\Rightarrow\neg\alpha)$
(D) $(\neg(\alpha\lor\beta))\equiv(\neg\alpha\Rightarrow\neg\beta)$
Choose the correct answer from the options given below:

Check statement (A):

$(\alpha\land(\beta\lor\gamma))\equiv((\alpha\land\beta)\lor(\alpha\land\gamma))$

This is distributive law of logic.

So, statement (A) is true.

Check statement (B):

$(\alpha\lor\beta)\equiv\neg\alpha\lor\beta$

This is not always true.

For example, let $\alpha=True$ and $\beta=False$.

Then,

$\alpha\lor\beta=True\lor False=True$

But,

$\neg\alpha\lor\beta=False\lor False=False$

Both values are different.

So, statement (B) is false.

Check statement (C):

$(\alpha\Rightarrow\beta)\equiv(\neg\beta\Rightarrow\neg\alpha)$

This is contrapositive law.

Also,

$\alpha\Rightarrow\beta\equiv\neg\alpha\lor\beta$

And,

$\neg\beta\Rightarrow\neg\alpha\equiv\beta\lor\neg\alpha$

Since,

$\neg\alpha\lor\beta\equiv\beta\lor\neg\alpha$

So, statement (C) is true.

Check statement (D):

$(\neg(\alpha\lor\beta))\equiv(\neg\alpha\Rightarrow\neg\beta)$

Using De Morgan's law,

$\neg(\alpha\lor\beta)\equiv\neg\alpha\land\neg\beta$

Now,

$\neg\alpha\Rightarrow\neg\beta\equiv\alpha\lor\neg\beta$

These two are not equivalent.

So, statement (D) is false.

Therefore, only (A) and (C) are correct.