Solution

Correct answer: (A) and (C) only

(A) (α ∧ (β ∨ γ)) ≡ ((α ∧ β) ∨ (α ∧ γ))
This logical equivalence says that the expression (α ∧ (β ∨ γ)) is logically equivalent to the expression ((α ∧ β) ∨ (α ∧ γ)). This can be thought of as the associative law of conjunction, which states that the order of the variables in conjunction does not affect the truth value of the conjunction. In other words, grouping the variables in a different way does not change the truth value of the expression.

(B) (α ∨ β) ≡ −α ∨ β
This logical equivalence says that the expression (α ∨ β) is logically equivalent to the expression −α ∨ β. However, this is not a correct equivalence because the negation of a logical variable (in this case α) is not equivalent to the logical variable itself. For example, if α is true, then −α is false, and therefore the two expressions have different truth values.

(C) (α ⇒ β) ≡ (¬β ⇒ ¬α)

This logical equivalence says that the expression (α ⇒ β) is logically equivalent to the expression (¬β ⇒ ¬α). This can be thought of as the contrapositive law, which states that the contrapositive of a logical implication is logically equivalent to the original implication. In other words, if we change the antecedent and consequent of a logical implication and negate both, the resulting expression is logically equivalent to the original implication.

(D) (¬(α ∨ β)) ≡ (¬α ⇒ ¬β)

This logical equivalence says that the expression (¬(α ∨ β)) is logically equivalent to the expression (¬α ⇒ ¬β). However, this is not a correct equivalence because the negation of a disjunction (α ∨ β) is not equivalent to a material implication (¬α ⇒ ¬β). For example, if α is true and β is false, then the expression (α ∨ β) is true and its negation (¬(α ∨ β)) is false. On the other hand, the expression (¬α ⇒ ¬β) is false because ¬α is false and ¬β is true, and therefore the two expressions have different truth values