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Let $\text{P}$ be a propositional variable.

Which of the following propositions are tautologies?

1. $\text{P}$
2. $\text{P} \Rightarrow \text{P}$
3. $(\text{P} \Rightarrow \text{P}) \Rightarrow \text{P}$
4. $\text{P} \Rightarrow(\text{P} \Rightarrow \text{P})$

1. $\text{P}$

$\text{Answer:}$ not a tautology

1. $\text{P} \Rightarrow \text{P}$

$\text{Answer:}$ tautology

1. $(\text{P} \Rightarrow \text{P}) \Rightarrow \text{P}$

$\text{Answer:}$ not a tautology.

$\text{Example reasoning:}$

Not all rows in the truth table evaluate to true.
$$\begin{array}{|c|c|c|}\hline \text{P} & \text{P} \Rightarrow \text{P} & (\text{P}\Rightarrow \text{P})\Rightarrow \text{P} \\ \hline \text{T} & \text{T} & \text{T} \\\hline \text{F} & \text{T} & \text{F} \\\hline \end{array}$$

1. $\text{P} \Rightarrow (\text{P} \Rightarrow \text{P})$

$\text{Answer:}$ tautology

Detailed Video Solution: https://youtu.be/nclBhBmtz2g?t=313