MATH661

1.Numbers

Exercise 1. Define one-to-one correspondences between the following sets of numbers:

$𝔼 = {n | n \in ℕ, n mod 2 = 0 .}$ , $𝕆 = {n | n \in ℕ, n mod 2 = 1 .}$

Solution. $f : 𝔼 \to 𝕆$ , $f (n) = n + 1$ is one-to-one.

$ℕ$ , $ℤ$

Solution. $f : ℕ \to ℤ$ , with $f$ defined by ${0 \to 0, 1 \to - 1, 2 \to 1, 3 \to - 2, 4 \to 2, \dots}$ is one possibility. Introducing $[x]$ as the integer part of $x \in ℚ$ , i.e. $[x] = n$ with $n ⩽ x < n + 1$ , $f$ can be expressed as

f (n) = {(- 1)}^{n mod 2} ([n / 2] + n mod 2)

In Julia $x \div y$ is integer division, so for $n \in ℕ$ $[n / 2] = n \div 2$ , and % is the modulo operator

∴	[4÷5 4÷4 4÷3 4÷2; 4%5 4%4 4%3 4%2]

$[\begin{array}{cccc} 0 & 1 & 1 & 2 \\ 4 & 0 & 1 & 0 \end{array}]$ (1)

∴	function f(n) q = n÷2; r = n%2; s = 1-2r; s(q+r) end

f

∴	N=10; [collect(0:N)'; f.(0:N)']

$[\begin{array}{ccccccccccc} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 0 & - 1 & 1 & - 2 & 2 & - 3 & 3 & - 4 & 4 & - 5 & 5 \end{array}]$ (2)

∴

$ℤ$ , $ℚ$

Solution. Construct a table and introduce diagonal traversal to obtain the positive rationals $p / q$ .

Figure 1.

From above deduce $| ℕ | = | 𝔼 | = | 𝕆 | = | ℤ | = | ℚ | = ℵ_{0}$ .

Exercise 2. Provide an example to show $| ℝ | > | ℕ |$ .

Exercise 3. Let $ℕ_{q} = {n | n . \in ℕ, n < 2^{q}}$ . Answer the following questions analytically. Also provide a Julia implementation.

Define a one-to-one correspondence $f : ℕ_{2 q} \to ℕ_{q} \times ℕ_{q}$
Let $f (m_{1}) = (n_{1}, p_{1})$ , $f (m_{2}) = (n_{2}, p_{2})$ . Assume $m_{1} + m_{2} \in ℕ_{2 q}$ . Express $f (m_{1} + m_{2})$ in terms of $n_{1}, n_{2}, p_{1}, p_{2}$ .
Assume $m_{1} \cdot m_{2} \in ℕ_{2 q}$ . Express $f (m_{1} \cdot m_{2})$ in terms of $n_{1}, n_{2}, p_{1}, p_{2}$ .

Exercise 4. Construct a one-to-one representation of the positions of atoms within an hexagonal lattice, $f : ℤ^{2} \to ℝ^{2}$ . Implement $f$ and $f^{- 1}$ as Julia functions. Use $f$ to construct a graphical representation of a two-dimensional hexagonal lattice.

Exercise 5. Construct a one-to-one representation of the positions of atoms within an hexagonal lattice, $f : ℤ^{3} \to ℝ^{3}$ . Implement $f$ and $f^{- 1}$ as Julia functions. Use $f$ to construct a graphical representation of a three-dimensional hexagonal lattice.

Number Approximation - Exercises

1.Numbers

2.Approximation

3.Successive approximations