Question

Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.​

Answer 1

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\begin{array}{c |c| c} \sf Value \: of \: x & \sf Value \: of \: y = 3x& \sf Whether \: x , \: Y \:€ \: A \\ \hline \sf 1 & \sf 3 \times 1 = 3 & \sf Yes \\ \hline \sf 2 & \sf 3 \times 2 = 6 & \sf Yes \\ \hline \sf 3 & \sf 3 \times 3 = 9 & \sf Yes \\ \hline \sf 4 & \sf 3 \times4 = 12 & \sf Yes \\ \hline \sf 5 & \sf 3 \times 5 = 15 & \sf No \\ \hline \sf 6 & \sf 3 \times 6 = 18 & \sf No \\ \hline \sf 7 & \sf 3 \times 7 = 21 & \sf No \end{array}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$

$\\\\\bf It \: is \: given \: that$

$~~~~~~~\implies ~~\sf \footnotesize 3x - y = 0$

$~~~~~~~\implies ~~\sf \footnotesize 3x = 0 + y$

$~~~~~~~\implies ~~\sf \footnotesize 3x = y$

$~~~~~~~\implies ~~\sf \footnotesize y = 3x$

Now,

$\sf {R = \{ ( 1 , 3 ) , ( 2 , 6 ) , ( 3,9 ) , ( 4 , 12 ) \}}$

$\bf \mathcal{Domain ~of~ R = Set ~of ~all~ first~ elements~ of ~the~ ordered~ pairs }$

$\sf ~~~~~~~~~~~~= { 1 , 2 , 3,4 }$

$\mathcal{Range~ of~ R = Set ~of ~all~ second~ elements~ of ~the~ ordered ~pairs}$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: = { 3 , 6 , 9 , 12 }$

$\bf {Codomain}$

$~~~~~~~~~~\mathcal{R~ is~ defined ~from ~A~ to~ A}$

$\sf\therefore Codomain \: of \: R = A$

$\sf\: \: \: \: \: \: \: \: \: \: \: \: = { 1 , 2 , 3 , ... , 14 }$

Answer 2

Step-by-step explanation:

It is given that

3x-y=0

3x = 0 + y

3x = y

y = 3x

$R={(1, 3), (2, 6), (3,9), (4, 12)}$

Domain of R = Set of all first elements of the ordered pairs

$= {1, 2, 3, 4)$

Range of R = Set of all second elements of the ordered pairs

$= {3, 6, 9, 12)$

Codomain

R is defined from A to A

R is defined from A to ACodomain of R = A= = {1, 2, 3, ..., 14)