(4) Every cockroach that is clever eats garbage. There is a cockroach that likes dirt or does not like dust. For each cockroach, it is not the case that it likes dirt or eats garbage. Therefore there is a cockroach such that it is not the case that if it is not clever then it likes dust.
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- Apply the pigeonhole principle to answer the following questions. If the pigeonhole principle cannot be applied, give a specific counterexample. 1. A team of three high jumpers all have a personal record that is at least 6 feet and less than 7 feet. Is it necessarily true that two of the team members must have personal records that are within four inches of each other? 2. What if there are four jumpers? Note that here heights are measured to within a precision of 1/4 inchWhich of the following statements is CORRECT? 1. b. The typical R2 for a stock is about 0.94 and the typical R2 for a portfolio is about 0.6. 2. c. The typical R2 for a stock is about 0.3 and the typical R2 for a large portfolio is about 0.94. 3. d. The typical R2 for a stock is about 0.94 and the typical R2 for a portfolio is also about 0. 4. e. The typical R2 for a stock is about 0.6 and the typical R2 for a portfolio is also about 0.6. 5. a. The typical R2 for a stock is about 0.3 and the typical R2 for a portfolio is also about 0.3.Adapt the argument in Example 2 to prove rigorously thatlim x 2 = c2 for all c. x→ c
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