Let r $$\in$$ {p, q, $$\sim$$p, $$\sim$$q} be such that the logical statement

r $$\vee$$ ($$\sim$$p) $$\Rightarrow$$ (p $$\wedge$$ q) $$\vee$$ r

is a tautology. Then r is equal to :

Let $$\Delta$$, $$\nabla $$ $$\in$$ {$$\wedge$$, $$\vee$$} be such that p $$\nabla$$ q $$\Rightarrow$$ ((p $$\Delta$$ q) $$\nabla$$ r) is a tautology. Then (p $$\nabla$$ q) $$\Delta$$ r is logically equivalent to :

The negation of the Boolean expression (($$\sim$$ q) $$\wedge$$ p) $$\Rightarrow$$ (($$\sim$$ p) $$\vee$$ q) is logically equivalent to :

Consider the following two propositions:

$$P1: \sim (p \to \sim q)$$

$$P2:(p \wedge \sim q) \wedge (( \sim p) \vee q)$$

If the proposition $$p \to (( \sim p) \vee q)$$ is evaluated as FALSE, then :