Chapter 5.4, Problem 6E

a.

To determine

To calculate:

Arrangement of cardinal numbers in order.

Answer to Problem 6E

The correct order is:

  $\overline{\overline{\varphi }}<\overline{\overline{\left\{0\right\}}}<\overline{\overline{\left\{0,1\right\}}}<\overline{\overline{\mathbb{Q}}}<\overline{\overline{\left(0,1\right)}}=\overline{\overline{ℝ-\mathbb{N}}}=\overline{\overline{ℝ}}<\overline{\overline{P\left(ℝ\right)}}<\overline{\overline{P\left(P\left(ℝ\right)\right)}}$

Explanation of Solution

Given information:

  $\overline{\overline{\left(0,1\right)}},\overline{\overline{\left\{0,1\right\}}},\overline{\overline{\left\{0\right\}}},\overline{\overline{P\left(ℝ\right)}},\overline{\overline{\mathbb{Q}}},\overline{\overline{\varphi }},\overline{\overline{ℝ-\mathbb{N}}},\overline{\overline{P\left(P\left(ℝ\right)\right)}},\overline{\overline{ℝ}}$

Calculation:

Consider the given cardinal numbers,

  $\overline{\overline{\left(0,1\right)}},\overline{\overline{\left\{0,1\right\}}},\overline{\overline{\left\{0\right\}}},\overline{\overline{P\left(ℝ\right)}},\overline{\overline{\mathbb{Q}}},\overline{\overline{\varphi }},\overline{\overline{ℝ-\mathbb{N}}},\overline{\overline{P\left(P\left(ℝ\right)\right)}},\overline{\overline{ℝ}}$

Arrange the following cardinal numbers in order.

The correct order is as below,

  $\overline{\overline{\varphi }}<\overline{\overline{\left\{0\right\}}}<\overline{\overline{\left\{0,1\right\}}}<\overline{\overline{\mathbb{Q}}}<\overline{\overline{\left(0,1\right)}}=\overline{\overline{ℝ-\mathbb{N}}}=\overline{\overline{ℝ}}<\overline{\overline{P\left(ℝ\right)}}<\overline{\overline{P\left(P\left(ℝ\right)\right)}}$

Since all intervals are equivalent.

b.

To determine

To calculate:

Arrangement of cardinal numbers in order.

Answer to Problem 6E

The correct order is:

  $\overline{\overline{\left\{0,5\right\}}}<\overline{\overline{\left\{0,3,5\right\}}}<\overline{\overline{P\left(\left\{0,5\right\}\right)}}<\overline{\overline{\left(0,5\right)-\left\{3\right\}}}=\overline{\overline{\left[0,5\right]}}=\overline{\overline{ℝ-\left\{3\right\}}}=\overline{\overline{ℝ-\mathbb{N}}}<\overline{\overline{P-\left(\left(0,5\right)\right)}}$

Explanation of Solution

Given information:

  $\overline{\overline{\left\{0,5\right\}}},\overline{\overline{\left[0,5\right]}},\overline{\overline{\left\{0,3,5\right\}}},\overline{\overline{ℝ-\left\{3\right\}}},\overline{\overline{P\left(\left\{0,5\right\}\right)}},\overline{\overline{P\left(\left(0,5\right)\right)}},\overline{\overline{\left(0,5\right)-\left\{3\right\}}},\overline{\overline{ℝ-\mathbb{N}}}$

Calculation:

Consider the given cardinal numbers,

  $\overline{\overline{\left\{0,5\right\}}},\overline{\overline{\left[0,5\right]}},\overline{\overline{\left\{0,3,5\right\}}},\overline{\overline{ℝ-\left\{3\right\}}},\overline{\overline{P\left(\left\{0,5\right\}\right)}},\overline{\overline{P\left(\left(0,5\right)\right)}},\overline{\overline{\left(0,5\right)-\left\{3\right\}}},\overline{\overline{ℝ-\mathbb{N}}}$

Arrange the following cardinal numbers in order.

The correct order is as below,

  $\overline{\overline{\left\{0,5\right\}}}<\overline{\overline{\left\{0,3,5\right\}}}<\overline{\overline{P\left(\left\{0,5\right\}\right)}}<\overline{\overline{\left(0,5\right)-\left\{3\right\}}}=\overline{\overline{\left[0,5\right]}}=\overline{\overline{ℝ-\left\{3\right\}}}=\overline{\overline{ℝ-\mathbb{N}}}<\overline{\overline{P-\left(\left(0,5\right)\right)}}$

Since all intervals are equivalent.

c.

To determine

To calculate:

Arrangement of cardinal numbers in order.

Answer to Problem 6E

The correct order is:

  $\overline{\overline{\left\{0,3,5\right\}}}<\overline{\overline{P\left(\left\{0,1\right\}\right)}}<\overline{\overline{\mathbb{Q}U\left\{\pi \right\}}}=\overline{\overline{\mathbb{Z}}}<\overline{\overline{\left[0,2\right]}}=\overline{\overline{\left(0,\infty \right)}}=\overline{\overline{ℝ-\left\{\pi \right\}}}=\overline{\overline{ℝ-\mathbb{Z}}}<\overline{\overline{P-\left(ℝ\right)}}$

Explanation of Solution

Given information:

  $\overline{\overline{\mathbb{Q}\cup \left\{\pi \right\}}},\overline{\overline{ℝ-\left\{\pi \right\}}},\overline{\overline{P\left(\left\{0,1\right\}\right)}},\overline{\overline{\left[0,2\right]}},\overline{\overline{\left(0,\infty \right)}},\overline{\overline{\mathbb{Z}}},\overline{\overline{ℝ-\mathbb{Z}}},\overline{\overline{P\left(ℝ\right)}},$

Calculation:

Consider the given cardinal numbers,

  $\overline{\overline{\mathbb{Q}\cup \left\{\pi \right\}}},\overline{\overline{ℝ-\left\{\pi \right\}}},\overline{\overline{P\left(\left\{0,1\right\}\right)}},\overline{\overline{\left[0,2\right]}},\overline{\overline{\left(0,\infty \right)}},\overline{\overline{\mathbb{Z}}},\overline{\overline{ℝ-\mathbb{Z}}},\overline{\overline{P\left(ℝ\right)}},$

Arrange the following cardinal numbers in order.

The correct order is as below,

  $\overline{\overline{\left\{0,3,5\right\}}}<\overline{\overline{P\left(\left\{0,1\right\}\right)}}<\overline{\overline{\mathbb{Q}U\left\{\pi \right\}}}=\overline{\overline{\mathbb{Z}}}<\overline{\overline{\left[0,2\right]}}=\overline{\overline{\left(0,\infty \right)}}=\overline{\overline{ℝ-\left\{\pi \right\}}}=\overline{\overline{ℝ-\mathbb{Z}}}<\overline{\overline{P-\left(ℝ\right)}}$

Since all intervals are equivalent.