Let S be an infinite set and S_1, S_2, ..., S_n be sets such that S_1 ∪ S_2 ∪ ... ∪ S_n = S. Then A. at least one of the sets S_i is a finite set B. not more than one of the sets S_i can be finite C. at least one of the sets S_i is an infinite set D. not more than one of the sets S_i can be infinite E. None of the above

GATE 1993 · Discrete Mathematics · Set Theory · medium

Answer: C. at least one of the sets S_i is an infinite set

1. Assume all S_i are finite — derive a contradiction: If each S_i is finite with |S_i| = k_i, then |S_1 ∪ ... ∪ S_n| <= k_1 + ... + k_n, a finite number.
2. Conclude at least one S_i must be infinite: But S is infinite, so |S_1 ∪ ... ∪ S_n| is infinite — contradicting the assumption. Hence at least one S_i must be infinite.