Comibnatorics question, parity , bijection recurrence

4. A paritypreserving subset {0:1, . . . ,ak} of {1, . . . ,n} satises the parity condition ai E i(mod 2), for i = 1, . . . , 1:. Let an be the number of parity-preserving subsets of {1, . . . ,n} (of any cardinality), n 2 0. (3) Using the differencepartial sum bijection, prove that n 1+:r, Zuni :1_ _ 2. \"20 :r a: (1)) From (a), we have the recurrence an 2 (15.1-1 + an_2, n 2 2, with initial conditions a0 = 1, a1 = 2. Prove this recurrence directly by describing a bijection between parity preserving subsets. (e) Let I)" be the number of subsets (of any cardinality) of {1, . . . ,n} with no consecutive pairs of elements, 71 2 0. Using the difference-partial sum bijection, prove that (d) From parts (a) and (c), we have proved that an 2 b\