We could think of a set union as a function that maps from several sets in A(the domain) to a set in B(the codomain). However, according to answers I have it an onto function. Could I get an explanation for the logic behind the reasoning in the answer?
- We can think of ∪ (set union) and ∩ (set intersection) as functions. What are the domain and codomain of ∪? Is ∪ a 1-to-1 function? Is ∪ an onto function? Answer: Let U be the universal set, and P(U) be the powerset of U.
• The domain of ∪ is P(U) × P(U) and the codomain is P(U).
• Let A be any nonempty subset of U. Then, (A, ∅) and (∅, A) are distinct elements in the domain, yet ∪ maps them both to the same element A in the codomain. Hence, ∪ is not 1-to-1.
• For every set A in the codomain P(U), ∪ maps (A, ∅) to A. Hence, ∪ is onto.