Question

Φ × A = φ for all A, where φ is the empty set.
true
false

Answer 1

The statement that the product of the empty set (Φ) with any set A equals the empty set (Φ) is true, as the Cartesian product with the empty set always results in the empty set.

Step-by-step explanation:

The statement Φ × A = Φ for all A, where Φ is the empty set, is true. In set theory, the Cartesian product of any set with the empty set results in the empty set. There are no elements in Φ to pair with the elements in set A, so the product of these two sets Φ × A is Φ itself.

This is an application of the properties of the empty set within set theory and applies to any set A. The statement that the product of the empty set (Φ) with any set A equals the empty set (Φ) is true, as the Cartesian product with the empty set always results in the empty set.

In mathematics, the equation φ × A = φ for all A, where φ represents the empty set, is true. When you multiply any set, including the empty set, by the empty set, the result is always the empty set.