MHB What is the Tautology in the Given Logical Equivalence?

[Lancelot1]

Hi guys

I can't figure this one out. I tried to use truth tables, but never found an equivalence , no matter which of the 5 options I tried.

It is given that $\alpha$ is logically equivalent to $\alpha \rightarrow \sim \beta $ .
Which of the following is a tautology ?

1) $\alpha$
2) $\beta$
3) $\alpha \wedge \beta $
4) $\beta \vee \sim\alpha $
5) $\alpha \leftrightarrow \beta $

 

[HOI]

You are correct. None of these is a tautology.
((3) is a contradiction- it is always false.)

 

[Evgeny.Makarov]

It is given that α\alpha is logically equivalent to $\alpha \rightarrow {\sim}\beta$.

I would say the problem with this exercise is that it is not clear what type of object $\alpha$ is. Is it a truth value? Then the question whether $\alpha$ is a tautology does not make sense. Is it a Boolean formula? Then it cannot be logically equivalent to $\alpha \rightarrow {\sim}\beta$. Indeed, if $\beta=1$, then $\alpha$ has to have the same value as ${\sim}\alpha$.