Question

Exercises 34–37 deal with these relations on the set of real numbers:

\({R_1} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a > b} \right\},\)the “greater than” relation,

\({R_2} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ge b} \right\},\)the “greater than or equal to” relation,

\({R_3} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a < b} \right\},\)the “less than” relation,

\({R_4} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \le b} \right\},\)the “less than or equal to” relation,

\({R_5} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a = b} \right\},\)the “equal to” relation,

\({R_6} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ne b} \right\},\)the “unequal to” relation.

36. Find

(a) \({R_1}^\circ {R_1}\).

(b) \({R_1}^\circ {R_2}\).

(c) \({R_1}^\circ {R_3}\).

(d) \({R_1}^\circ {R_4}\).

(e) \({R_1}^\circ {R_5}\).

(f) \({R_1}^\circ {R_6}\).

(g) \({R_2}^\circ {R_3}\).

(h) \({R_3}^\circ {R_3}\).

Short answer

(a)The solution of Relation\({R_1} \circ {R_1} = {R_1}.\)

(b) The solution of Relation\({R_1} \circ {R_2} = {R_1}\)

(c) The solution of Relation\({R_1} \circ {R_3} = {\mathbb{R}^2}\)

(d) The solution of Relation\({R_1}^\circ {R_4} = {\mathbb{R}^2}\)

(e) The solution of Relation\({R_1}^\circ {R_5} = {R_1}\)

(f) The solution of Relation\({R_1}^\circ {R_6} = {\mathbb{R}^2}\)

(g) The solution of Relation\({R_2}^\circ {R_3} = {\mathbb{R}^2}\)

(h) The solution of Relation \({R_3}^\circ {R_3} = {R_3}\)

Step-by-step solution

  Given

The given for all parts are as follows:

\(\begin{array}{l}{R_1} = \left\{ {(a,b) \in {R^2}/a > b} \right\}\\{R_2} = \left\{ {(a,b) \in {R^2}/a \ge b} \right\}\\{R_3} = \left\{ {(a,b) \in {R^2}/a < b} \right\},\\{R_4} = \left\{ {(a,b) \in {R^2}/a \le b} \right\}\\{R_5} = \left\{ {(a,b) \in {R^2}/a = b} \right\},\\{R_6} = \left\{ {(a,b) \in {R^2}/a \ne b} \right\}\end{array}\)

The Concept of relation

An n-array relation on n sets, is any subset of Cartesian product of the n sets (i.e., a collection ofn-tuples), with the most common one being a binary relation, a collection of order pairs from two sets containing an object from each set. The relation is homogeneous when it is formed with one set.

(a) Determine the value of relation

For every\((a,b) \in R\)and\((b,c) \in S\), the corresponding element\((a,c)\)belongs to the composition.

\(\begin{array}{c}{R_1}^\circ {R_1} = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_1} \Rightarrow b{\rm{ such that }}(a,b) \in {R_1}{\rm{ and }}(b,c) \in {R_1}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_1} \Rightarrow a > b,b > c \Leftrightarrow a > c} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_1} \Rightarrow (a,c) \in {R_1}} \right\}\\{R_1}^\circ {R_1} = {R_1}\end{array}\)

(b) Determine the value of relation

For every\((a,b) \in R\)and\((b,c) \in S\), the corresponding element\((a,c)\)belongs to the composition.

\(\begin{array}{c}{R_1}^\circ {R_2} = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_2} \Rightarrow b{\rm{ such that }}(a,b) \in {R_1}{\rm{ and }}(b,c) \in {R_2}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_2} \Rightarrow a > b,b \ge c \Leftrightarrow a > c} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_2} \Rightarrow (a,c) \in {R_1}} \right\}\\{R_1}^\circ {R_2} = {R_1}\end{array}\)

(c) Determine the value of relation

For every\((a,b) \in R\)and\((b,c) \in S\), the corresponding element\((a,c)\)belongs to the composition.

\(\begin{array}{c}{R_1}^\circ {R_3} = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_3} \Rightarrow b{\rm{ such that }}(a,b) \in {R_3}{\rm{ and }}(b,c) \in {R_1}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_3} \Rightarrow a < b,b > c \Leftrightarrow b{\rm{ is high }}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_3} \Rightarrow (a,c) \in {\mathbb{R}^2}} \right\}\\{R_1}^\circ {R_3} = {\mathbb{R}^2}\end{array}\)

(d) Determine the value of relation

For every\((a,b) \in R\)and\((b,c) \in S\), the corresponding element\((a,c)\)belongs to the composition.

\(\begin{array}{c}{R_1}^\circ {R_4} = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_4} \Rightarrow b{\rm{ such that }}(a,b) \in {R_1}{\rm{ and }}(b,c) \in {R_4}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_4} \Rightarrow a > b,b \le c \Leftrightarrow b{\rm{ is less }}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_3} \Rightarrow (a,c) \in {\mathbb{R}^2}} \right\}\end{array}\)

(e) Determine the value of relation

For every\((a,b) \in R\)and\((b,c) \in S\), the corresponding element\((a,c)\)belongs to the composition.

\(\begin{array}{c}{R_1}^\circ {R_5} = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_5} \Rightarrow b{\rm{ such that }}(a,b) \in {R_1}{\rm{ and }}(b,c) \in {R_5}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_5} \Rightarrow a > b,b = c \Leftrightarrow a > c} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_5} \Rightarrow (a,c) \in {R_1}} \right\}\\{R_1}^\circ {R_5} = {R_1}\end{array}\)

(f) Determine the value of relation

For every\((a,b) \in R\)and\((b,c) \in S\), the corresponding element ( a, c) belongs to the composition.

\(\begin{array}{c}{R_1}^\circ {R_6} = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_6} \Rightarrow b{\rm{ such that }}(a,b) \in {R_1}{\rm{ and }}(b,c) \in {R_6}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_6} \Rightarrow a > b,b \ne c \Leftrightarrow b{\rm{ is less }}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_1}^\circ {R_6} \Rightarrow (a,c) \in {\mathbb{R}^2}} \right\}\\{R_1}^\circ {R_6} = {\mathbb{R}^2}\end{array}\)

(g) Determine the value of relation

For every\((a,b) \in R\)and\((b,c) \in S\), the corresponding element\((a,c)\)belongs to the composition.

\(\begin{array}{c}{R_2}^\circ {R_3} = \left\{ {(a,b)/(a,c) \in {R_2}^\circ {R_3} \Rightarrow b{\rm{ such that }}(a,b) \in {R_2}{\rm{ and }}(b,c) \in {R_3}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_2}^\circ {R_3} \Rightarrow a \ge b,b < c \Leftrightarrow b{\rm{ is less }}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_2}^\circ {R_3} \Rightarrow (a,c) \in {\mathbb{R}^2}} \right\}\\{R_2}^\circ {R_3} = {\mathbb{R}^2}\end{array}\)

(h) Determine the value of relation

The solution of Relation

For every \((a,b) \in R\) and \((b,c) \in S\), the corresponding element \((a,c)\) belongs to the composition.

\(\begin{array}{c}{R_3}^\circ {R_3} = \left\{ {(a,b)/(a,c) \in {R_3}^\circ {R_3} \Rightarrow b{\rm{ such that }}(a,b) \in {R_3}{\rm{ and }}(b,c) \in {R_3}} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_3}^\circ {R_3} \Rightarrow a < b,b < c \Leftrightarrow a < c} \right\}\\ = \left\{ {(a,b)/(a,c) \in {R_2}^\circ {R_3} \Rightarrow (a,c) \in {R_3}} \right\}\\{R_3}^\circ {R_3} = {R_3}\end{array}\)