Question Number 22652 by Tinkutara last updated on 21/Oct/17
![Find the number of unordered pairs {A, B} (i.e., the pairs {A, B} and {B, A} are considered to be the same) of subsets of an n-element set X which satisfy the conditions: (a) A ≠ B; (b) A ∪ B = X [e.g., if X = {a, b, c, d}, then {{a, b}, {b, c, d}}, {{a}, {b, c, d}}, {φ, {a, b, c, d}} are some of the admissible pairs.]](Q22652.png)
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{unordered}\:\mathrm{pairs} \\ $$$$\left\{{A},\:{B}\right\}\:\left(\mathrm{i}.\mathrm{e}.,\:\mathrm{the}\:\mathrm{pairs}\:\left\{{A},\:{B}\right\}\:\mathrm{and}\:\left\{{B},\:{A}\right\}\right. \\ $$$$\left.\mathrm{are}\:\mathrm{considered}\:\mathrm{to}\:\mathrm{be}\:\mathrm{the}\:\mathrm{same}\right)\:\mathrm{of} \\ $$$$\mathrm{subsets}\:\mathrm{of}\:\mathrm{an}\:{n}-\mathrm{element}\:\mathrm{set}\:{X}\:\mathrm{which} \\ $$$$\mathrm{satisfy}\:\mathrm{the}\:\mathrm{conditions}: \\ $$$$\left(\mathrm{a}\right)\:{A}\:\neq\:{B}; \\ $$$$\left(\mathrm{b}\right)\:{A}\:\cup\:{B}\:=\:{X} \\ $$$$\left[\mathrm{e}.\mathrm{g}.,\:\mathrm{if}\:{X}\:=\:\left\{{a},\:{b},\:{c},\:{d}\right\},\:\mathrm{then}\:\left\{\left\{{a},\:{b}\right\},\right.\right. \\ $$$$\left.\left\{{b},\:{c},\:{d}\right\}\right\},\:\left\{\left\{{a}\right\},\:\left\{{b},\:{c},\:{d}\right\}\right\},\:\left\{\phi,\:\left\{{a},\:{b},\:{c},\:{d}\right\}\right\} \\ $$$$\left.\mathrm{are}\:\mathrm{some}\:\mathrm{of}\:\mathrm{the}\:\mathrm{admissible}\:\mathrm{pairs}.\right] \\ $$