Question

Suppose \(B \neq \varnothing\) and \(A \times B \subseteq B \times C .\) Prove that \(A \subseteq C\).

Short answer

The claim \(A \subseteq C\) is true. A key conceptual understanding here was of the definitions of Cartesian products and subsets: if for all (a, b) in A × B, there is a (a, c) in B × C, then all elements a in A are also in C.

Step-by-step solution

Understand the Problem and Notations

First, understand the terms: `B ≠ ∅` means set B is not an empty set. `A × B ⊆ B × C` means that the Cartesian product of A and B is a subset of the Cartesian product of B and C. The Cartesian product of two sets, say A and B, (notated as A × B), is the set of all ordered pairs (a,b) such that a is in A and b is in B. `A ⊆ C` means all elements of A are also elements of C, which needs to be proven.

Assume an Element in A

For this a proof, take an arbitrary element a in A. Since B is non-empty, there exists at least one element b in B.

Arrive at the Conclusion

By the property of Cartesian products, for A × B to be a subset of B × C, for every (a, b) in A × B, there must be a corresponding pair in B × C. This means that for our arbitrary a in A, and some b in B, since (a, b) would be an element of A × B, there must exist (a, c) in B × C, for some c in C. As such, it can be concluded that all elements of A, including our chosen element a, are in C.