Final answer:
The function theta defined on P(Z) is bijective as it is both injective (one-to-one) and surjective (onto), with each subset mapping uniquely to itself. The inverse of this function is the function itself, since theta(X) = X for any subset X of integers.
Step-by-step explanation:
The question is asking whether the function theta, defined on the power set of integers P(Z), and given by theta(X) = X for any subset X of the integers, is bijective. To determine if theta is bijective, we need to check if it is both injective (one-to-one) and surjective (onto).
A function is injective if different elements in the domain map to different elements in the codomain. Here, if X and Y are two different subsets of the integers, then theta(X) will be X and theta(Y) will be Y and since X is different from Y, their images under theta are also different. Therefore, theta is injective.
A function is surjective if every element in the codomain is the image of at least one element in the domain. In this case, for any subset Y of the integers, there is a subset X in the domain such that theta(X) = X = Y. Hence, theta is surjective.
Since theta is both injective and surjective, it is indeed bijective. The inverse of a bijective function reverses the mapping. For the function theta, the inverse is theta itself because theta-1(Y) = Y. Therefore, the inverse of theta is also theta.