Which of the following propositions is a tautology0/1p ∧ ¬𝑝p ∨ 𝑝 (p ∧ ¬𝑝) ∨ (p ∨ ¬𝑝)(p ∨ ¬𝑝) → (p ∧ ¬𝑝)

Sure, let's go through each proposition one by one:

1. p ∧ ¬𝑝: This is a contradiction, not a tautology. It says that p is both true and not true at the same time, which is impossible.

2. p ∨ 𝑝: This is a tautology. It says that either p is true, or p is true. So, no matter what the truth value of p is, the whole proposition is true.

3. (p ∧ ¬𝑝) ∨ (p ∨ ¬𝑝): This is a tautology. The first part (p ∧ ¬𝑝) is a contradiction, but the second part (p ∨ ¬𝑝) is a tautology. Since we have "or" between them, if either part is true, the whole proposition is true. So, this proposition is always true.

4. (p ∨ ¬𝑝) → (p ∧ ¬𝑝): This is not a tautology. The first part (p ∨ ¬𝑝) is a tautology, but the second part (p ∧ ¬𝑝) is a contradiction. The implication is only false when the first part is true and the second part is false, which is the case here. So, this proposition is not always true.

So, the tautologies are the second and the third propositions.