Final answer:
The question deals with mathematical functions and the selection of subset pairs (B, C) and subsets (E) in relation with function images and pre-images. By giving examples of these subsets, we can demonstrate scenarios where the properties of function composition and inversion reveal interesting behaviors in terms of set equality.
Step-by-step explanation:
The question involves functions in mathematics, specifically understanding the concept of a function image and pre-image. The question asks for two subsets, B and C, of the real numbers R such that the property f(C\B) ≠ f(C) \ f(B) holds. This can be illustrated by choosing specific subsets where the function f(x) = x2 maps different elements from C to the same element in f(C), which could be removed by f(B). An example of B and C satisfying this could be B = {1} and C = {-1, 1}, where f(C\B) includes the image of -1, but f(C) would not have it since it is removed by f(B).
Additionally, the question asks to find a subset E such that f-1(f(E)) ≠ E. An example of such a set is E = {-1, 1}, since f-1(f(E)) would yield the set {-1, 1} as f(E) would be {1}, and the pre-image would return both -1 and 1, despite -1 not being in the original set.