In 41–49 let X and Y be sets, let A and B be any subsets of X, and let C and D be any subsets of Y . Determine which of the properties are true for all functions F from X to Y and which are false for at least one function F from X to Y . Justify your answers. 47. For all subsets C and D of Y , F−1 (C ∩ D) = F−1 (C) ∩ F−1 (D).

The property F−1(C ∩ D) = F−1(C) ∩ F−1(D) is true for all functions F from set X to set Y according to the Preimage of Intersection Property for functions. The proof relies on the concept that any element in the preimage of the intersection of two subsets must belong to the preimages of both subsets, establishing equality.

Explanation:

The property mentioned, F−1(C ∩ D) = F−1(C) ∩ F−1(D), is true for all functions F from set X to set Y. This is known as the Preimage of Intersection Property for functions.

To justify this, let's take an arbitrary element x that belongs to the preimage of the intersection of sets C and D. This means that its image under function F lies in the intersection of C and D, so it belongs to both sets C and D. Hence, x belongs to the preimage of C and to the preimage of D. So, x is an element of the intersection of the preimages of C and D. This shows that the preimage of the intersection is a subset of the intersection of the preimages.

The reverse inclusion can be proved similarly, which establishes the equality.

Learn more about Preimage of Intersection Property here:

https://brainly.com/question/35348361

#SPJ11