Final answer:
The given options are:
a. f is injective but not surjective.
b. f is surjective but not injective.
c. X = Y and f is injective but not surjective.
d. [X] = [Y] and f is surjective but not injective.
e. [X] = [Y]. X and Y are finite, and f is injective but not surjective.
f. [X] = [Y], X and Y are finite, and f is surjective but not injective.
Options a, b, c, d, e, and f are all possible.
Step-by-step explanation:
Let's analyze these options:
a. If a function is injective, it means that each element in the domain (X) has a unique element in the codomain (Y).
However, it does not mean that each element in the codomain is mapped from an element in the domain.
Therefore, option a is possible.
b. If a function is surjective, it means that each element in the codomain (Y) has at least one element in the domain (X) that maps to it.
However, it does not mean that each element in the domain has a unique element in the codomain.
Therefore, option b is possible.
c. If X = Y and the function is injective but not surjective, it means that each element in X has a unique element in X that maps to it, but there are elements in Y that are not mapped from an element in X.
Therefore, option c is possible.
d. If [X] = [Y] and the function is surjective but not injective, it means that every element in X and Y has at least one corresponding element in the other set, but there are elements in X that are not uniquely mapped to an element in Y.
Therefore, option d is possible.
e. If [X] = [Y] and X, Y are finite, and the function is injective but not surjective, it means that each element in X has a unique element in X that maps to it, but not every element in Y is mapped from an element in X.
Therefore, option e is possible.
f. If [X] = [Y] and X, Y are finite, and the function is surjective but not injective, it means that every element in X and Y has at least one corresponding element in the other set, but there are elements in X that are not uniquely mapped to an element in Y.
Therefore, option f is possible.