Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums, and Matrices

Question: Find $\mathit{f}\mathbf{}\mathbf{+}\mathbf{}\mathit{g}$and$\mathit{f}\mathit{g}$ for the functions f and g given in exercise 36.

Sum both sides of the identity${\mathrm{k}}^2-(\mathrm{k}-1{)}^2=2\mathrm{k}-1\text{from}\mathrm{k}=1\text{to}\mathrm{k}=\mathrm{n}$ to and use Exercise 35 to find

a) a formula for$\sum _{k=1}^n (2k-1)$ (the sum of the first n odd natural numbers).

b) a formula for $\sum _{k=1}^n \mathrm{k}$

Find$A^n$ if A is$\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$

Show that \(A \times B \ne B \times A\), when A and B are non empty, unless \(A = B\)

Show that if A and B are sets, then

a) \(A \oplus B{\bf{ = }}B \oplus A\)

b) \(\left( {A \oplus B} \right) \oplus B = A\)

Question: Let\(f(x) = ax + b\) and \(g(x) = cx + d\) where a, b, c, and d are constants. Determine necessary and sufficient conditions on the constants a, b, c, and d so that \(f \circ g = g \circ f\)

Question: A pair of dice is rolled in a remote location and when you ask an honest observer whether at least one die came up six, this honest observer answers in the affirmative.

a) What is the probability that the sum of the numbers that came up on the two dice is seven, given the information provided by the honest observer?

b) Suppose that the honest observer tells us that at least one die came up five. What is the probability the sum of the numbers that came up on the dice is seven, given this information?

Use the technique given in Exercise 35, together with the result of Exercise , to derive the formula for$\sum _{k=1}^n k^2$ given in Table 2. [Hint: Take${\mathrm{a}}_{\mathrm{k}}={\mathrm{k}}^3$ in the telescoping sum in Exercise 35.]

Let$f\left(x\right)=ax+b$ and $g\left(x\right)=cx+d$where a, b, c, and d are constants. Determine necessary and sufficient conditions on the constants a, b, c, and d so that$f\circ g=g\circ f$

Show that the set of functions from the positive integers to the set{1,2,3,4,5,6,7,8,9,} is uncountable. [Hint:First set up a one-to-one correspondence between the set of real numbers between 0 and 1 and a subset of these functions. Do this by associating to the real number 0 . $d_1{d}_2...d_n.......$ the function f with $f\left(n\right)=d_n$ .