Question

Suppose A1 and A2 are affine subsets of V. Prove that the intersection A1 A2 is either:

A) An empty set
B) An infinite set
C) An affine subset of V
D) A finite set

Answer 1

The intersection of two affine subsets A1 and A2 is an affine subset of V because it satisfies the properties of an affine subset.

Step-by-step explanation:

To prove that the intersection of two affine subsets A1 and A2 is an affine subset of V, we need to show that the intersection satisfies the properties of an affine subset.

Let x, y be any two points in the intersection (A1 ∩ A2). Since x and y belong to both A1 and A2, they satisfy the defining equations of both subsets.

Therefore, any linear combination of x and y will also satisfy the defining equations of both A1 and A2, making it an element of the intersection. This shows that (A1 ∩ A2) is closed under linear combinations and hence, it is an affine subset of V.