1. Consider a function f : A → B. Fill in the blank with a statement relating |A| and |B|, and then prove the resulting claim: if , then, for some b ∈ B, we have | {a ∈ A : f(a) = b} | ≥ 202.   2....

1. Consider a function f : A → B. Fill in the blank with a statement relating |A| and |B|, and then prove the resulting claim: if , then, for some b ∈ B, we have | {a ∈ A : f(a) = b} | ≥ 202.

2. Suppose that we quantize a set of values from S = {1, 2, . . . , n} into {k1, k2, . . . , k5} ⊂ S. (See Example 2.56.) Namely, we choose these 5 values and then define a function q : S → {k1, k2, . . . , k5}. The maximum error of this quantization is maxx∈S |x − q(x)|. Use the Pigeonhole Principle (or the “the maximum must exceed the average” generalization) to determine the smallest possible maximum error.