Which of the following statements are logically equivalent?

(A) No liquids are beverages.

(B) No beverages are liquids.

(C) All non-beverages are non-liquids.

(D) All non-liquids are non-beverages.

Choose the correct answer from the options given below:

The correct answer is (A) and (B) Only.

To determine which statements are logically equivalent, we can analyze each categorical proposition and represent it using set theory or formal logic.

Let L represent the class of "liquids" and B represent the class of "beverages". Let L^c and B^c denote their complements ("non-liquids" and "non-beverages", respectively).

1. Analysis of Statement (A): "No liquids are beverages."
This is a universal negative proposition (E-proposition). It states that the set of liquids and the set of beverages have no elements in common (they are disjoint sets).
Mathematically, we can write this as:
$$L\cap B=\varnothing$$
This means that there is no overlap between the two categories.

2. Analysis of Statement (B): "No beverages are liquids."
This is also a universal negative proposition (E-proposition). It states that the set of beverages and the set of liquids are disjoint.
Mathematically, this is expressed as:
$$B\cap L=\varnothing$$
Since set intersection is commutative:
$$L\cap B=B\cap L$$
Statement (A) and Statement (B) are logically equivalent. In formal logic, this is justified by the rule of simple conversion, which states that a universal negative (E) proposition can be validly converted by swapping the subject and predicate terms.

3. Analysis of Statement (C): "All non-beverages are non-liquids."
This is a universal affirmative proposition (A-proposition) relating the complements: "All B^c are L^c".
In terms of subsets, this is written as:
$$B^c\subseteq L^c$$
By the logical rule of contraposition, the statement "All B^c are L^c" is logically equivalent to:
$$L\subseteq B$$
Which translates to: "All liquids are beverages." This describes a relation of inclusion, which is not equivalent to the exclusion stated in (A) and (B).

4. Analysis of Statement (D): "All non-liquids are non-beverages."
This statement asserts: "All L^c are B^c".
In set notation, this is:
$$L^c\subseteq B^c$$
By contraposition, this is logically equivalent to:
$$B\subseteq L$$
Which translates to: "All beverages are liquids." This is also an inclusion relation and not equivalent to (A) or (B).

Conclusion:
Only statements (A) and (B) share the exact same logical meaning of mutual exclusion between liquids and beverages.