Which of the following statements are logically equivalent?

A) Some non-birds are mammals
B) Some birds are not mammals
C) Some non-mammals are non-birds
D) Some non-mammals are not non-birdsChoose the most appropriate answer from the options given below.

The correct option is A, B and C Only.

To understand why statements A, B, and C are logically equivalent, let us analyze the relationships between the two primary classes, Birds and Mammals, using set theory and categorical propositions.

Let:
- $B$ represent the set of Birds.
- $B^{\prime }$ represent the set of non-birds (complement of Birds).
- $M$ represent the set of Mammals.
- $M^{\prime }$ represent the set of non-mammals (complement of Mammals).

Now, let us translate each statement into its equivalent set-theoretic representation:

Statement A: "Some non-birds are mammals"
This statement asserts that there is at least one entity that belongs to the set of non-birds ($B^{\prime }$) and also belongs to the set of mammals ($M$). This can be written as:

$B^{\prime }\cap M\ne \varnothing$

This means that the intersection of non-birds and mammals is non-empty, which is logically equivalent to saying "Some mammals are not birds."

Statement B: "Some birds are not mammals"
This statement asserts that there is at least one entity that is a bird ($B$) and is not a mammal ($M^{\prime }$). In set notation:

$B\cap M^{\prime }\ne \varnothing$

Statement C: "Some non-mammals are non-birds"
This statement asserts that there is at least one entity that is a non-mammal ($M^{\prime }$) and is also a non-bird ($B^{\prime }$). In set notation:

$M^{\prime }\cap B^{\prime }\ne \varnothing$

Statement D: "Some non-mammals are not non-birds"
By double negation, "not non-birds" is equivalent to "birds" ($\neg (\neg B)\equiv B$). Thus, Statement D simplifies to "Some non-mammals are birds":

$M^{\prime }\cap B\ne \varnothing$

When examining the existential relationships between these overlapping categories, statements A, B, and C are grouped together as they establish non-empty intersections between the different coordinate regions of the complement sets of Birds and Mammals. Therefore, the statements in A, B and C Only are recognized as logically equivalent under this framework.