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

Show that there is a one-to-one correspondence from the set of subsets of the positive integers to the set real numbers between0 and 1 . Use this result and Exercises 34 and 35 to conclude that $N_0<\left|p\left(Z^{+}\right)\left|=\right|R\right|$. [Hint:Look at the first part of the hint for Exercise 35 .]

Question: Find $\mathit{f}\mathbf{\circ }\mathit{g}$and $\mathit{g}\mathbf{\circ }\mathit{f}$, where$\mathbf{}\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$ and $\mathit{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}$, are functions from $\mathbf{ℝ}\mathbf{}\mathit{t}\mathit{o}\mathbf{ℝ}$.

How many different elements does A X B X C have if A has m elements, B has n elements and C has p elements?

Use the identity$\frac{1}{\left(\mathrm{k}\right(\mathrm{k}+1\left)\right)}=\frac{\frac{1}{\mathrm{k}-1}}{(\mathrm{k}+1)}$ and Exercise 35 to compute$\sum _{k=1}^n \frac{1}{\left(k\right(k+1\left)\right)}$

Question:To determine which is more likely: rolling a total of 8 when two dice are rolled or rolling a total of 8 when three dice are rolled.

Show that C, the set of complex numbers has the same cardinality as R, the set of real numbers.

How many different element does \({{\bf{A}}^{\bf{n}}}\), have when A has m elements and n is a positive integer.

Show that if A is a subset of a universal set U, then

Find f + g and fg for the functions f and g given in exercise 36.

Show that the set of all computer programs in a particular programming language is countable. [Hint: A computer program written in a programming language can be thought of as a string of symbols from a finite alphabet.]