Final answer:
The cardinality function c(A) and product function p(A) are both functions. c(A) is injective and would be onto with a codomain of {1, 2, 3}. p(A) is injective and would be onto with a codomain of {1, 2, 4, 8}. s(A) is not traditionally considered a function.
Step-by-step explanation:
Let us consider set S as the set of all non-empty subsets of the interval [1, 2, 4]. As per the student's question, we need to determine whether the given operations are functions and, if they are, whether they are one-to-one (injective) and onto (surjective), plus the appropriate codomain. a) Cardinality Function c(A) = |A|. For any set A ∈ S, c(A) represents the number of elements in A. This is indeed a function because it assigns exactly one output (the size of set A) to each input (set A). It is injective because different subsets will have different cardinalities (except for sets with the same size).
The codomain that makes this function possible would be the set of positive integers that are possible sizes of subsets of [1, 2, 4]. In this case, the codomain is {1, 2, 3}. b) Subset Elements Function s(A). The function s(A) returns the set of all elements in the set A ∈ S. This is not a function in the conventional sense, as a function in set theory assigns a single output to each input, and here the output is a subset of the original set, so it differs from the standard definition of a function. c) Product Function p(A) The function p(A) computes the product of elements in a set A ∈ S. This is a function because it assigns one output (the product) to each input set. It is injective if we only consider non-empty subsets, since the product of the elements uniquely identifies the subset. The codomain making it onto (or surjective) would be the set of all possible products you could get from elements of [1, 2, 4], which are {1, 2, 4, 8}.