Final answer:
The question pertains to Cartesian products and subset relations in set theory. A x B is a subset of A x B x C because each element of A x B can be paired with an element of C to form an element of A x B x C.
Step-by-step explanation:
The question asks if A x B is a subset of A x B x C given that sets A, B, C are not empty. To address this, we will review the nature of Cartesian products and subset relations.
Cartesian products involve pairing each element of one set with every element of another set. The Cartesian product A x B consists of all ordered pairs (a, b) where a is an element of A and b is an element of B. When we extend this to a third set C, creating A x B x C, it involves ordered triples (a, b, c), where a is an element of A, b is an element of B, and c is an element of C.
An ordered pair (a, b) from A x B can be associated with an arbitrary element c of C to form an ordered triple (a, b, c) within A x B x C. Since this can be done for all elements in A x B, it implies that every element of A x B is included in A x B x C, therefore A x B is indeed a subset of A x B x C.