91Ó°ÊÓ

Suppose six pairs of similar-looking boots are thrown together in a pile. How many individual boots must you pick to be sure of getting a matched pair? Why?

How many integers from 0 through 60 must you pick in order to be sure of getting at least one that is odd? at least one that is even?

Show that the set of all bit strings (strings of 0 's and 1's) is countable.

Use the definition of logarithm to fill in the blanks below. a. \(\log _{2} 8=3\) because b. \(\log _{5}\left(\frac{1}{25}\right)-2\) because c. \(\log _{4} 4=1\) because d. \(\log _{3}\left(3^{n}\right)=n\) because e. \(\log _{4} 1=0\) because

How many integers from 1 through 100 must you pick in order to be sure of getting one that is divisible by 5 ?

How many integers must you pick in order to be sure that at least two of them have the same remainder when divided by \(7 ?\)

Must the average of two irrational numbers always be irrational? Prove or give a counterexample.

In each of 16-19 a function \(f\) is defined on a set of real numbers. Determine whether or not \(f\) is one-to-one and justify your answer. $$ f(x)=\frac{3 x-1}{x}, \text { for all real numbers } x \neq 0 $$

How many integers must you pick in order to be sure that at leasi two of them have the same remainder when divided by \(15 ?\)

How many integers from 100 through 999 must you pick in order to be sure that at least two of them have a digit in common? (For example, 256 and 530 have the common digit \(5 .)\)