Answer Happy

Solve these problems:
12. Evaluate each of the following A. Let A, B be sets. Prove that if |AU BI = A[ + \BI, then An B=0. - B. Let A, B be sets. Prove that (A - B) (B - A) = 0. C. Let A, B be non-empty sets. Prove that if AXB = BXA, then A = B. D. Prove that in any set of n numbers, there is one number whose value is at least the average of the n numbers. E. Let A, B be finite sets. Prove that if A - B = 0 and there is a bijection between A and B, then A = B. F. This problem is taken from Maryland Math Olympiad problem, and was posted on the Computational Complexity Web Log. Suppose we color each of the natural numbers with a color from {red, blue, green}. Prove that there exist distinct x, y such that (x - yl is a perfect square. (Hint: it suffices to consider the integers between o and 225). V- G. Prove that 3 is irrational. One way to do this is similar to the proof done in class that 2 is irrational, but consider two cases depending on whether a is even or odd.