91Ó°ÊÓ

a. Given any set of seven integers, must there be two that have the same remainder when divided by 6 ? Why? b. Given any set of seven integers, must there be two that have the same remainder when divided by \(8 ?\) Why?

Let \(S\) be the set of all strings in \(a\) 's and \(b\) 's and let \(L: S \rightarrow \mathbf{Z}\) be the length function: For all strings \(s \in S\), $$ L(s)=\text { the number of characters in } s \text {. } $$ Let \(T: \mathbf{Z} \rightarrow\\{0,1,2\\}\) be the \(\bmod 3\) function: For all integers \(n, \quad T(n)=n \bmod 3 .\) What is \((T \circ L)(a b a a) ?(T \circ L)(b a a a b) ?(T \circ L)(a a a) ?\)

Show that the set of all nonnegative integers is countable by exhibiting a one-to-one correspondence between \(\mathbf{Z}^{+}\)and \(\mathbf{Z}^{\text {nonneg }}\).

a. If seven integers are chosen from between 1 and 12 inclusive, must at least one of them be odd? Why? b. If ten integers are chosen from between 1 and 20 inclusive, must at least one of them be even? Why?

If \(n+1\) integers are chosen from the set $$ \\{1,2,3, \ldots, 2 n\\} $$ where \(n\) is a positive integer, must at least one of them be odd? Why?

a. How many one-to-one functions are there from a set with three elements to a set with four elements? b. How many one-to-one functions are there from a set with three elements to a set with two elements? c. How many one-to-one functions are there from a set with three elements to a set with three elements? d. How many one-to-one functions are there from a set with three elements to a set with five elements? e. How many one-to-one functions are there from a set with \(m\) elements to a set with \(n\) elements, where \(m \leq n\) ?

a. How many onto functions are there from a set with three elements to a set with two elements? b. How many onto functions are there from a set with three elements to a set with five elements? c. How many onto functions are there from a set with three elements to a set with three elements? d. How many onto functions are there from a set with four elements to a set with two elements? e. How many onto functions are there from a set with four elements to a set with three elements?

The functions of each pair in \(9-11\) are inverse to each other. For each pair, check that both compositions give the identity function. \(H\) and \(H^{-1}\) are both defined from \(\mathbf{R}-\\{1\\}\) to \(\mathbf{R}-\\{1\\}\) by the formula \(H(x)=H^{-1}(x)=\frac{x+1}{x-1}, \quad\) for all \(x \in \mathbf{R}-\\{1\\} .\)

How many cards must you pick from a standard 52 -card deck to be sure of getting at least I red card? Why?

a. Define \(f: \mathbf{Z} \rightarrow \mathbf{Z}\) by the rule \(f(n)=2 n\), for all integers \(n\). (i) Is \(f\) one-to-one? Prove or give a counterexample. (ii) Is \(f\) onto? Prove or give a counterexample. b. Let \(2 \mathbf{Z}\) denote the set of all even integers. That is, \(2 \mathbf{Z}=\) \(\\{n \in \mathbf{Z} \mid n=2 k\), for some integer \(k\\}\). Define \(h: \mathbf{Z} \rightarrow 2 \mathbf{Z}\) by the rule \(h(n)=2 n\), for all integers \(n\). Is \(h\) onto? Prove or give a counterexample.