Chapter 3.5, Problem 2E

a.

To determine

To give: an example of the relation that is antisymmetric and symmetric.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

Using the definition, the relation which is antisymmetric and symmetric is as follows

  $R_1=\left\{\left(a,a\right),\left(b,b\right),\left(c,c\right)\right\}$

Clearly the relation is antisymmetric and symmetric.

b.

To determine

To give an example of the relation that is antisymmetric, reflexive on A and not symmetric.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

The relation is reflexive if $\left(a,a\right)\in R$

Using the definition, the relation which is antisymmetric and reflexive on A but not symmetric is as follows

  $R_2=\left\{\left(a,a\right),\left(b,b\right),\left(c,c\right),\left(a,b\right)\right\}$

Clearly the relation is antisymmetric.

The relation is reflexive as $\left(a,a\right)\in R_2$ , $\left(b,b\right)\in R_2$

  $\left(c,c\right)\in R_2$ .

The relation is not symmetric as $\left(a,b\right)\in R_2$ but $\left(b,a\right)\notin R_2$

c.

To determine

To give an example of the relation that is antisymmetric, not reflexive on A and not symmetric.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

The relation is reflexive if $\left(a,a\right)\in R$

Using the definition, the relation that is antisymmetric, not reflexive on A and not symmetric is as follows

  $R_3=\left\{\left(a,a\right),\left(b,a\right)\right\}$

Clearly the relation is antisymmetric.

The relation is not reflexive as $\left(b,b\right)\notin R_3$ .

The relation is not symmetric as $\left(b,a\right)\in R_3$ but $\left(a,b\right)\notin R_3$

d.

To determine

To give an example of the relation that symmetric and not antisymmetric.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

Using the definition, the relation that is symmetric and not antisymmetric is as follows

  $R_4=\left\{\left(a,b\right),\left(b,a\right)\right\}$

Clearly the relation is not anti-symmetric as $a\ne b$

The relation is symmetric as $\left(a,b\right)\in R_4$ and $\left(b,a\right)\in R_4$

e.

To determine

To give an example of the relation that isnot symmetric and not antisymmetric.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

Using the definition, the relation that is not symmetric and not antisymmetric is as follows

  $R_5=\left\{\left(a,b\right),\left(b,a\right),\left(a,c\right)\right\}$

Clearly the relation is not anti-symmetric as $a\ne b$

The relation is not symmetric as $\left(a,c\right)\in R_5$ but $\left(c,a\right)\in R_5$

f.

To determine

To give an example of the relation that is irreflexive on A and not symmetric.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$

Using the definition, the relation that irreflexive on A and not symmetric is as follows

  $R_6=\left\{\left(b,a\right),\left(a,c\right)\right\}$

The relation is ir-reflexive as $\left(c,c\right),\left(b,b\right),\left(a,a\right)\notin R_6$

The relation is not symmetric as $\left(b,a\right)\in R_6$ , $\left(a,c\right)\in R_6$ but $\left(c,a\right)\notin R_6$ and $\left(a,b\right)\notin R_6$ .

g.

To determine

To give an example of the relation that is irreflexive on A and not antisymmetric.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

Using the definition, the relation that is irreflexive on A and not antisymmetric is as follows

  $R_7=\left\{\left(b,a\right),\left(a,b\right)\right\}$

The relation is irreflexive as $\left(c,c\right),\left(b,b\right),\left(a,a\right)\notin R_6$

Also, the relation, $\left(b,a\right)\in R_7$ , $\left(a,b\right)\in R_7$ but $a\ne b$ . Therefore, the relation is not anti-symmetric.

h.

To determine

To give an example of the relation that is antisymmetric, not reflexive and irreflexive on A.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

Using the definition, the relation that is antisymmetric, not reflexive and irreflexive on A is as follows

  $R_8=\left\{\left(a,a\right)\right\}$

The relation is irreflexive as $\left(c,c\right),\left(b,b\right)\notin R_8$

i.

To determine

To give an example of the relation that is transitive, antisymmetric and irreflexive on A.

Explanation of Solution

Given Information: The set $A=\{a,b,c\}$

The relation is anti-symmetric if $\left(a,b\right)\in R$ and $\left(b,a\right)\in R$ then $a=b$

The relation is transitive if $\left(a,b\right)\in R$ and $\left(b,c\right)\in R$ then $\left(a,c\right)\in R$

Using the definition, the relation that transitive, antisymmetric and irreflexive on A.is as follows

  $R_9=\left\{\left(a,b\right),\left(b,a\right)\left(a,a\right)\right\}$

The relation is transitiveas $\left(a,b\right)\in R_9$ and $\left(b,a\right)\in R_9$ then $\left(a,a\right)\in R_9$

The relation is irreflexive as $\left(c,c\right),\left(b,b\right)\notin R_9$