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infinite family of functions

Source: miklos schweitzer 1997 q6

September 24, 2021
infinityfunction

Problem Statement

Let κ\kappa be an infinite cardinality and let A , B be sets of cardinality κ\kappa. Construct a family F\cal F of functions f:A→Bf : A \to B with cardinality 2κ2^\kappa such that for all functions f1,⋯ ,fn∈Ff_1,\cdots, f_n \in\cal F and for all b1,...,bn∈Bb_1 , ..., b_n \in B, there exist a∈Aa\in A such that f1(a)=b1,⋯ ,fn(a)=bnf_1(a) = b_1,\cdots, f_n(a) = b_n.
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