If f : X -> Y and a, b ⊆ X, then f (a ∩ b) is equal toans.f(b) - f(a)a proper subset of f(a) ∩ f(b)f(a) ∩ f(b)f(a) - f(b)

The correct answer is f(a) ∩ f(b).

Here's why:

The function f : X -> Y maps elements from set X to set Y. If a and b are subsets of X, then f(a) and f(b) are the images of these subsets in Y under the function f.

The intersection of two sets, denoted by '∩', is a set that contains all elements that are common to both sets. So, a ∩ b is the set of all elements that are in both a and b.

When we apply the function f to this intersection, i.e., f(a ∩ b), we get the set of all images in Y of the elements that are common to both a and b.

On the other hand, f(a) ∩ f(b) is the set of all elements in Y that are common to both f(a) and f(b).

Since f(a ∩ b) is the set of all images of the elements common to a and b, and f(a) ∩ f(b) is the set of all common elements of f(a) and f(b), it follows that f(a ∩ b) is a subset of f(a) ∩ f(b).

However, in general, f(a ∩ b) is not necessarily equal to f(a) ∩ f(b). This is because there might be elements in f(a) ∩ f(b) that are not in f(a ∩ b). This happens when the function f is not injective, i.e., when different elements in X can have the same image in Y.

So, the correct answer is that f(a ∩ b) is a subset of f(a) ∩ f(b), but not necessarily equal to it.