The proposition is true because the disjunction of two propositions is true if at least one of the propositions is true. In this case, either P or Q is true, so the proposition is true.

The truth table for the proposition is as follows:

The proposition ∼(P ∨ Q) is logically equivalent to ∉(P ∨ Q) because the negation of a disjunction is equivalent to the conjunction of the negations of the propositions.

The truth table for the proposition is as follows:

The proposition (P ∨ Q) ∧ (∉P ∧ ∉Q) is logically equivalent to ∼(P ∧ Q) because the conjunction of two propositions is false if at least one of the propositions is false.

The truth table for the proposition is as follows:

The proposition (∉P ∨ ∉Q) ∧ (P ∨ Q) is logically equivalent to ∼(P ∨ Q) because the conjunction of two propositions is false if at least one of the propositions is false.

The truth table for the proposition is as follows:

The proposition (P ∧ Q) ∨ (P ∨ ∉Q) is logically equivalent to P ∧ Q because the conjunction of two propositions is true if both propositions are true.

The truth table for the proposition is as follows:

The proposition (∉P ∨ ∉Q) ∨ (P ∧ ∉Q) is logically equivalent to ∼(P ∨ Q) because the conjunction of two propositions is false if at least one of the propositions is false.

The truth table for the proposition is as follows:

The proposition (P ∨ Q) ∧ (∉P ∨ Q) is logically equivalent to ∼(P ∧ Q) because the conjunction of two propositions is false if at least one of the propositions is false.