MathDB

4

Part of 2005 ISI B.Math Entrance Exam

Problems(1)

interesting cardinality

Source:

5/5/2012
For a set SS we denote its cardinality by ∣S∣|S|. Let e1,e2,…,eke_1,e_2,\ldots,e_k be non-negative integers. Let AkA_k (respectively BkB_k) be the set of all kk-tuples (f1,f2,…,fk)(f_1,f_2,\ldots,f_k) of integers such that 0≤fi≤ei0\leq f_i\leq e_i for all ii and ∑i=1kfi\sum_{i=1}^k f_i is even (respectively odd). Show that ∣Ak∣−∣Bk∣=0 or 1|A_k|-|B_k|=0 \textrm{ or } 1.
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