The statement $( eg p) \Rightarrow ( eg q)$ is logically equivalent to which of the statements below? I. $p \Rightarrow q$ II. $q \Rightarrow p$ III. $( eg q) \vee p$ IV. $( eg p) \vee q$

To determine logical equivalence, we can rewrite the conditional statement $( eg p) \Rightarrow ( eg q)$ using standard logical identities:

1. Disjunction form of conditional statement: $$(

eg p) \Rightarrow ( eg q) \equiveg(eg p) \vee ( eg q) \equiv p \vee ( eg q) \equiv ( eg q) \vee p$$ This matches statement III.

2. Contrapositive law: The contrapositive of $A \Rightarrow B$ is $

eg B \Rightarrow eg A$, which is logically equivalent to$A \Rightarrow B$. Applying contraposition to$( eg p) \Rightarrow ( eg q)$:$$eg(eg q) \Rightarroweg(eg p) \equiv q \Rightarrow p$$ This matches statement II.

Statements I and IV are logically equivalent to $p \Rightarrow q$ (or $eg p \vee q$), which is the inverse/converse rather than equivalent to $( eg p) \Rightarrow ( eg q)$.

Therefore, statements II and III only are logically equivalent to the given statement, which corresponds to option D.