Chapter 3.3, Problem 9E

(a)

To determine

To find: Relation on set A corresponding to the partition $\text{P}=\left\{\right\{1,2\},\{3,4,5\left\}\right\}$.

Answer to Problem 9E

  $\text{ R ={}(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(3,5),(4,3),(4,4),(4,5),(5,3),(5,4),(5,5)\}$

Explanation of Solution

Given Information:

  $\text{Let the given set be }\mathrm{A}=\{1,2,3,4,5\}$ .

Formula Used:

The ordered pair of two sets A and B is given as

  $\mathrm{A}\times \mathrm{B}=\{(\mathrm{a},\mathrm{b}):\mathrm{a}\in \mathrm{A},\mathrm{b}\in \mathrm{B}\}$

If $\text{{}\overline{{\mathrm{x}}_1},\overline{{\mathrm{x}}_2},\mathrm{...},\overline{{\mathrm{x}}_{\mathrm{n}}}\}$ gives the partition of set A for some relation then $\overline{{\mathrm{x}}_1}\times \overline{{\mathrm{x}}_1}\cup \overline{{\mathrm{x}}_2}\times \overline{{\mathrm{x}}_2}\cup \mathrm{...}\cup \overline{{\mathrm{x}}_{\mathrm{n}}}\times \overline{{\mathrm{x}}_{\mathrm{n}}}$ gives the corresponding relation.

Calculation:

Consider the given partition.

  $\text{P}=\left\{\right\{1,2\},\{3,4,5\left\}\right\}$

Calculate $\overline{{\mathrm{x}}_{\mathrm{n}}}\times \overline{{\mathrm{x}}_{\mathrm{n}}}$ for $\mathrm{n}=1,2$ as shown:

  $\begin{array}{l}\overline{{\mathrm{x}}_1}=\{1,2\}\\ \overline{{\mathrm{x}}_1}\times \overline{{\mathrm{x}}_1}=\{(1,1),(1,2),(2,1),(2,2)\}\\ \overline{{\mathrm{x}}_2}=\{3,4,5\}\\ \overline{{\mathrm{x}}_2}\times \overline{{\mathrm{x}}_2}=\{(3,3),(3,4),(3,5),(4,3),(4,4),(4,5),(5,3),(5,4),(5,5)\}\end{array}$

The corresponding relation can be found as

  $\begin{array}{l}\mathrm{R}\text{ =}\left(\overline{{\mathrm{x}}_1}\times \overline{{\mathrm{x}}_1}\right)\cup \left(\overline{{\mathrm{x}}_2}\times \overline{{\mathrm{x}}_2}\right)\\ \text{ =}\{(1,1),(1,2),(2,1),(2,2)\}\cup \{(3,3),(3,4),(3,5),(4,3),(4,4),(4,5),(5,3),(5,4),(5,5)\}\\ \text{ R ={}(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(3,5),(4,3),(4,4),(4,5),(5,3),(5,4),(5,5)\}\end{array}$

(b)

To determine

To find: Relation on set A corresponding to the partition $\text{ P}=\text{{{1},{2},{3,4},{5}}}$.

.

Answer to Problem 9E

  $\mathrm{R}=\{(1,1),(2,2),(3,3),(3,4),(4,3),(4,4),(5,5)\}$

Explanation of Solution

Given Information:

  $\text{Let the given set be }\mathrm{A}=\{1,2,3,4,5\}$ .

Formula Used:

The ordered pair of two sets A and B is given as

  $\mathrm{A}\times \mathrm{B}=\{(\mathrm{a},\mathrm{b}):\mathrm{a}\in \mathrm{A},\mathrm{b}\in \mathrm{B}\}$

If $\text{{}\overline{{\mathrm{x}}_1},\overline{{\mathrm{x}}_2},\mathrm{...},\overline{{\mathrm{x}}_{\mathrm{n}}}\}$ gives the partition of set A for some relation then $\overline{{\mathrm{x}}_1}\times \overline{{\mathrm{x}}_1}\cup \overline{{\mathrm{x}}_2}\times \overline{{\mathrm{x}}_2}\cup \mathrm{...}\cup \overline{{\mathrm{x}}_{\mathrm{n}}}\times \overline{{\mathrm{x}}_{\mathrm{n}}}$ gives the corresponding relation.

Calculation:

Consider the given partition.

  $\text{ P}=\text{{{1},{2},{3,4},{5}}}$

The corresponding relation can be found as

  $\begin{array}{l}\mathrm{R}=\left(\left\{1\right\}\times \left\{1\right\}\right)\cup \left(\left\{2\right\}\times \left\{2\right\}\right)\cup \left(\{3,4\}\times \{3,4\}\right)\cup \left(\left\{5\right\}\times \left\{5\right\}\right)\\ =\{(1,1),(2,2),(3,3),(3,4),(4,3),(4,4),(5,5)\}\end{array}$

(c)

To determine

To find: Relation on set A corresponding to the partition $\text{ P}=\text{{{2,3,4,5},{1}}}$.

Answer to Problem 9E

  $\begin{array}{l}\mathrm{R}=\text{{(2,2),(2,3),(2,4),(2,5),(3,2),(3,3),(3,4),(3,5),}\\ \text{ (4,2),(4,3),(4,4),(4,5),(5,2),(5,3),(5,4),(5,5),(1,1)}}\end{array}$

Explanation of Solution

Given Information:

  $\text{Let the given set be }\mathrm{A}=\{1,2,3,4,5\}$

Formula Used:

The ordered pair of two sets A and B is given as

  $\mathrm{A}\times \mathrm{B}=\{(\mathrm{a},\mathrm{b}):\mathrm{a}\in \mathrm{A},\mathrm{b}\in \mathrm{B}\}$

If $\text{{}\overline{{\mathrm{x}}_1},\overline{{\mathrm{x}}_2},\mathrm{...},\overline{{\mathrm{x}}_{\mathrm{n}}}\}$ gives the partition of set A for some relation then $\overline{{\mathrm{x}}_1}\times \overline{{\mathrm{x}}_1}\cup \overline{{\mathrm{x}}_2}\times \overline{{\mathrm{x}}_2}\cup \mathrm{...}\cup \overline{{\mathrm{x}}_{\mathrm{n}}}\times \overline{{\mathrm{x}}_{\mathrm{n}}}$ gives the corresponding relation.

Calculation:

Consider the given partition.

  $\text{ P}=\text{{{2,3,4,5},{1}}}$

The corresponding relation can be found as

  $\begin{array}{l}\text{R}=\left(\{2,3,4,5\}\times \text{{2,3,4,5}}\right)\cup \left(\text{{1}}\times \text{{1}}\right)\\ \text{ ={(2,2),(2,3),(2,4),(2,5),(3,2),(3,3),(3,4),(3,5),}\\ \text{ (4,2),(4,3),(4,4),(4,5),(5,2),(5,3),(5,4),(5,5),(1,1)}}\end{array}$

(d)

To determine

To find: Relation on set A corresponding to the partition $\text{P}=\text{{{1,5},{2,4},{3}}}$.

Answer to Problem 9E

  $\mathrm{R}=\{(1,1),(1,5),(5,1),(5,5),(2,2),(2,4),(4,2),(4,4),(3,3)\}$

Explanation of Solution

Given Information:

  $\text{Let the given set be }\mathrm{A}=\{1,2,3,4,5\}$

Formula Used:

The ordered pair of two sets A and B is given as

  $\mathrm{A}\times \mathrm{B}=\{(\mathrm{a},\mathrm{b}):\mathrm{a}\in \mathrm{A},\mathrm{b}\in \mathrm{B}\}$

If $\text{{}\overline{{\mathrm{x}}_1},\overline{{\mathrm{x}}_2},\mathrm{...},\overline{{\mathrm{x}}_{\mathrm{n}}}\}$ gives the partition of set A for some relation then $\overline{{\mathrm{x}}_1}\times \overline{{\mathrm{x}}_1}\cup \overline{{\mathrm{x}}_2}\times \overline{{\mathrm{x}}_2}\cup \mathrm{...}\cup \overline{{\mathrm{x}}_{\mathrm{n}}}\times \overline{{\mathrm{x}}_{\mathrm{n}}}$ gives the corresponding relation.

Calculation:

Consider the given partition.

  $\text{P}=\text{{{1,5},{2,4},{3}}}$

The corresponding relation can be found as

  $\begin{array}{l}\text{R}=\left(\text{{1,5}}\times \text{{1,5}}\right)\cup \left(\text{{2,4}}\times \text{{2,4}}\right)\cup \left(\text{{3}}\times \text{{3}}\right)\\ =\{(1,1),(1,5),(5,1),(5,5),(2,2),(2,4),(4,2),(4,4),(3,3)\}\end{array}$