Chapter 3.1, Problem 7E

a)

To determine

To find: the composite function $\mathrm{R}\circ \mathrm{S}$ .

Answer to Problem 7E

  $\mathrm{R}\circ \mathrm{S}=\left\{\left(3,5\right),\left(5,2\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

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

Here in the pair $\left(1,5\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{R}\right)=\left\{1,2,3,5\right\}$ and $\text{Range}\left(\mathrm{R}\right)=\left\{2,4,5\right\}$

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

Consider the set S .

  $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$

Here in the pair $\left(2,4\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{S}\right)=\left\{2,3,5\right\}$ and $\text{Range}\left(\mathrm{S}\right)=\left\{4,1,5\right\}$

  $\mathrm{S}\left(2\right)=4,\mathrm{S}\left(3\right)=4,\mathrm{S}\left(3\right)=1$ , and $\mathrm{S}\left(5\right)=4$

Construct the following diagram for the relation.

  

Hence, $\mathrm{R}\circ \mathrm{S}=\left\{\left(3,5\right),\left(5,2\right)\right\}$ .

b)

To determine

To find: the composite function $\mathrm{R}\circ \mathrm{T}$ .

Answer to Problem 7E

  $\mathrm{R}\circ \mathrm{T}=\left\{\left(3,2\right),\left(4,5\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

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

Here in the pair $\left(1,5\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{R}\right)=\left\{1,2,3,5\right\}$ and $\text{Range}\left(\mathrm{R}\right)=\left\{2,4,5\right\}$

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

Consider the set T .

  $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

  $\text{Dom}\left(\mathrm{T}\right)=\left\{1,3,4\right\}$ and $\text{Range}\left(\mathrm{T}\right)=\left\{4,5,1\right\}$

  $\mathrm{T}\left(1\right)=4,\mathrm{T}\left(3\right)=5$ , and $\mathrm{T}\left(4\right)=1$

Construct the following diagram for the relation.

  

Hence, $\mathrm{R}\circ \mathrm{T}=\left\{\left(3,2\right),\left(4,5\right)\right\}$ .

(c)

To determine

To find: the composite function $\mathrm{T}\circ \mathrm{S}$ .

Answer to Problem 7E

  $\mathrm{T}\circ \mathrm{S}=\left\{\left(2,1\right),\left(3,4\right),\left(3,1\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set S .

  $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$

Here in the pair $\left(2,4\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{S}\right)=\left\{2,3,5\right\}$ and $\text{Range}\left(\mathrm{S}\right)=\left\{4,1,5\right\}$

  $\mathrm{S}\left(2\right)=4,\mathrm{S}\left(3\right)=4,\mathrm{S}\left(3\right)=1$ , and $\mathrm{S}\left(5\right)=4$

Consider the set T .

  $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

  $\text{Dom}\left(\mathrm{T}\right)=\left\{1,3,4\right\}$ and $\text{Range}\left(\mathrm{T}\right)=\left\{4,5,1\right\}$

  $\mathrm{T}\left(1\right)=4,\mathrm{T}\left(3\right)=5$ , and $\mathrm{T}\left(4\right)=1$

Construct the following diagram for the relation.

  

Hence, $\mathrm{T}\circ \mathrm{S}=\left\{\left(2,1\right),\left(3,4\right),\left(3,1\right)\right\}$ .

(d)

To determine

To find: the composite function $\mathrm{R}\circ \mathrm{R}$ .

Answer to Problem 7E

  $\mathrm{R}\circ \mathrm{R}=\left\{\left(1,2\right),\left(2,2\right),\left(5,2\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

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

Here in the pair $\left(1,5\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{R}\right)=\left\{1,2,3,5\right\}$ and $\text{Range}\left(\mathrm{R}\right)=\left\{2,4,5\right\}$

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

Construct the following diagram for the relation.

  

Hence, $\mathrm{R}\circ \mathrm{R}=\left\{\left(1,2\right),\left(2,2\right),\left(5,2\right)\right\}$ .

(e)

To determine

To find: the composite function $\mathrm{S}\circ \mathrm{R}$ .

Answer to Problem 7E

  $\mathrm{S}\circ \mathrm{R}=\left\{\left(1,5\right),\left(2,4\right),\left(5,4\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

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

Here in the pair $\left(1,5\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{R}\right)=\left\{1,2,3,5\right\}$ and $\text{Range}\left(\mathrm{R}\right)=\left\{2,4,5\right\}$

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

Consider the set S .

  $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$

Here in the pair $\left(2,4\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{S}\right)=\left\{2,3,5\right\}$ and $\text{Range}\left(\mathrm{S}\right)=\left\{4,1,5\right\}$

  $\mathrm{S}\left(2\right)=4,\mathrm{S}\left(3\right)=4,\mathrm{S}\left(3\right)=1$ , and $\mathrm{S}\left(5\right)=4$

Construct the following diagram for the relation.

  

Hence, $\mathrm{S}\circ \mathrm{R}=\left\{\left(1,5\right),\left(2,4\right),\left(5,4\right)\right\}$ .

(f)

To determine

To find: the composite function $\mathrm{T}\circ \mathrm{T}$ .

Answer to Problem 7E

  $\mathrm{T}\circ \mathrm{T}=\left\{\left(1,1\right),\left(4,4\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set T .

  $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

  $\text{Dom}\left(\mathrm{T}\right)=\left\{1,3,4\right\}$ and $\text{Range}\left(\mathrm{T}\right)=\left\{4,5,1\right\}$

  $\mathrm{T}\left(1\right)=4,\mathrm{T}\left(3\right)=5$ , and $\mathrm{T}\left(4\right)=1$

Construct the following diagram for the relation.

  

Hence, $\mathrm{T}\circ \mathrm{T}=\left\{\left(1,1\right),\left(4,4\right)\right\}$ .

(g)

To determine

To find: the composite function $\mathrm{R}\circ \left(\mathrm{S}\circ \mathrm{T}\right)$ .

Answer to Problem 7E

  $\mathrm{R}\circ \left(\mathrm{S}\circ \mathrm{T}\right)=\left\{\left(3,2\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

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

Here in the pair $\left(1,5\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{R}\right)=\left\{1,2,3,5\right\}$ and $\text{Range}\left(\mathrm{R}\right)=\left\{2,4,5\right\}$

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

Consider the set S .

  $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$

Here in the pair $\left(2,4\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{S}\right)=\left\{2,3,5\right\}$ and $\text{Range}\left(\mathrm{S}\right)=\left\{4,1,5\right\}$

  $\mathrm{S}\left(2\right)=4,\mathrm{S}\left(3\right)=4,\mathrm{S}\left(3\right)=1$ , and $\mathrm{S}\left(5\right)=4$

Consider the set T .

  $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

  $\text{Dom}\left(\mathrm{T}\right)=\left\{1,3,4\right\}$ and $\text{Range}\left(\mathrm{T}\right)=\left\{4,5,1\right\}$

  $\mathrm{T}\left(1\right)=4,\mathrm{T}\left(3\right)=5$ , and $\mathrm{T}\left(4\right)=1$

Construct the following diagram for the relation.

  

  $\mathrm{S}\circ \mathrm{T}=\left\{\left(3,5\right)\right\}$

Now, find $\mathrm{R}\circ \left(\mathrm{S}\circ \mathrm{T}\right)$ as shown:

  $\left(\mathrm{S}\circ \mathrm{T}\right)\left(3\right)=5$ , $\mathrm{R}\left(1\right)=5,\mathrm{R}\left(2\right)=2,\mathrm{R}\left(3\right)=4$ , and $\mathrm{R}\left(5\right)=2$

Construct the following diagram for the relation.

  

Hence, $\mathrm{R}\circ \left(\mathrm{S}\circ \mathrm{T}\right)=\left\{\left(3,2\right)\right\}$ .

(h)

To determine

To find: the composite function $\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}$ .

Answer to Problem 7E

  $\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}=\left\{\left(3,2\right)\right\}$

Explanation of Solution

Given Information:

Three sets are given as

  $\mathrm{R}=\left\{\left(1,5\right),\left(2,2\right),\left(3,4\right),\left(5,2\right)\right\}$ , $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$ and $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

Formula used:

Composition function of two sets R and S can be evaluated as

  $\mathrm{R}\circ \mathrm{S}\left(\mathrm{x}\right)=\mathrm{R}\left[\mathrm{S}\left(\mathrm{x}\right)\right]=\mathrm{R}\left(\mathrm{y}\right)$

where x denote the domain of S and y denote the domain of R

Composition function of three sets R , S , and T can be evaluated as

  $\begin{array}{c}\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}\left(\mathrm{x}\right)=\mathrm{R}\circ \mathrm{S}\left[\mathrm{T}\left(\mathrm{x}\right)\right]\\ =\mathrm{R}\circ \mathrm{S}\left(\mathrm{y}\right)\\ =\mathrm{R}\left[\mathrm{S}\left(\mathrm{y}\right)\right]\\ =\mathrm{R}\left(\mathrm{z}\right)\end{array}$

where x is the domain of T and y is the domain of S and z is the domain of R

Calculation:

Consider the set R .

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

Here in the pair $\left(1,5\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{R}\right)=\left\{1,2,3,5\right\}$ and $\text{Range}\left(\mathrm{R}\right)=\left\{2,4,5\right\}$

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

Consider the set S .

  $\mathrm{S}=\left\{\left(2,4\right),\left(3,4\right),\left(3,1\right),\left(5,5\right)\right\}$

Here in the pair $\left(2,4\right)$ first element is in domain and second element is in range.

  $\text{Dom}\left(\mathrm{S}\right)=\left\{2,3,5\right\}$ and $\text{Range}\left(\mathrm{S}\right)=\left\{4,1,5\right\}$

  $\mathrm{S}\left(2\right)=4,\mathrm{S}\left(3\right)=4,\mathrm{S}\left(3\right)=1$ , and $\mathrm{S}\left(5\right)=4$

Consider the set T .

  $\mathrm{T}=\left\{\left(1,4\right),\left(3,5\right),\left(4,1\right)\right\}$

  $\text{Dom}\left(\mathrm{T}\right)=\left\{1,3,4\right\}$ and $\text{Range}\left(\mathrm{T}\right)=\left\{4,5,1\right\}$

  $\mathrm{T}\left(1\right)=4,\mathrm{T}\left(3\right)=5$ , and $\mathrm{T}\left(4\right)=1$

Construct the following diagram for the relation.

  

Hence, $\mathrm{R}\circ \mathrm{S}=\left\{\left(3,5\right),\left(5,2\right)\right\}$ .

Now, find $\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}$ as shown:

  $\left(\mathrm{R}\circ \mathrm{S}\right)\left(3\right)=5,\left(\mathrm{R}\circ \mathrm{S}\right)\left(5\right)=2$ , $\mathrm{T}\left(1\right)=4,\mathrm{T}\left(3\right)=5$ , and $\mathrm{T}\left(4\right)=1$

Construct the following diagram for the relation.

  

Hence, $\left(\mathrm{R}\circ \mathrm{S}\right)\circ \mathrm{T}=\left\{\left(3,2\right)\right\}$ .