you mean because the set of all functions from X to Y, its cardinal equals the number of possible mappings from X to Y, which is |Y|^|X|, right?matt grime said:Though the result you want to prove in the second post follows from counting the elements directly.
A set is a collection of distinct objects, called elements, which can be anything from numbers to letters to other sets.
Two sets are equal if and only if they contain the same elements. This means that if every element in one set is also in the other set, and vice versa, then the two sets are equal.
The union of two sets is the set of all elements that are in either set. This can be represented using the symbol "∪". For example, if set A = {1, 2, 3} and set B = {2, 4, 6}, then A ∪ B = {1, 2, 3, 4, 6}.
The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. This can be represented using the symbol "×". For example, if set A = {1, 2} and set B = {3, 4}, then A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.
To prove this equality, we need to show that every element in Y^(X∪{x}) is also in (Y^X)×(Y^{x}) and vice versa. This can be done by using the definition of the power set and the Cartesian product, as well as the fact that the union of two sets includes all elements from both sets.