Class Problem
Math 2513
Wednesday, July 6
Problem. Let Abe a set with nelements which are labelled a1, a2, . . . , an. If Bis a subset of Alet
f(B) be the bit string of length nwhich has a 1 in the ith position if aiโBand has a 0 in the ith
position if ai/โB. This defines a function ffrom the power set of Ato the set Bnconsisting of all
bit strings of length n. That is f:P โ Bn.
(a) In the case where n= 8, determine each of the following:
f(โ
), f(A), f({a5}), f({a8}) and f({a1, a3, a8}).
(b) In the case where n= 8, describe the subsets Bfor which f(B) is each of:
10101010,01010101,11110000,and 00001111.
(c) How can the cardinality of a subset Bbe determined by examining its corresponding bit string
f(B)?
ANSWERS:
(a) f(โ
) = 00000000, f(A) = 11111111, f({a5}) = 00001000, f({a8}) = 00000001 and f({a1, a3, a8}=
10100001.
(b) f({a1, a3, a5, a7}) = 10101010, f({a2, a4, a6, a8}= 01010101, f({a1, a2, a3, a4}= 11110000 , and
f({a5, a6, a7, a8}= 00001111.
(c) The cardinality of a subset Bof Aequals the number of 1โs in the bit string f(B).
