Question

a relation R on the set A={0,2,3,4,5,6,8,10,12} is defined as R={(x ,y):x, y belongs to A and 3x+ 4y =24} write the elation as a set ordered pairs, also find the domain and range of the relation

Answer 1

• The relation as a set ordered pairs is R = {(0,6) , (4,3) , (8,0)}

• The domain of the relation R = {0, 4, 8}

• The range of the relation R = {0, 3, 6}

Given :

A relation R on the set A = { 0 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 12 } is defined as R = {(x,y) : x,y belongs to A and 3x + 4y = 24 }

To find :

• The relation as a set ordered pairs

• The domain of the relation

• The range of the relation

Solution :

Step 1 of 4 :

Write down the given relation

A relation R on the set A = { 0 , 2 , 3 , 4 , 5 , 6 , 8 , 10 , 12 } is defined as

R = {(x,y) : x, y belongs to A and 3x + 4y = 24 }

Step 2 of 4 :

Express the relation as a set ordered pairs

$\displaystyle \sf{ 3x + 4y = 24}$

$\displaystyle \sf{ \implies y = \frac{24 - 3x}{4} }$

For x = 0 ∈ A we have y = 6 ∈ A

For x = 2 ∈ A we have y = 9/2 ∉ A

For x = 3 ∈ A we have y = 15/4 ∉ A

For x = 4 ∈ A we have y = 3 ∈ A

For x = 5 ∈ A we have y = 9/4 ∈ A

For x = 6 ∈ A we have y = 3/2 ∉ A

For x = 8 ∈ A we have y = 0 ∈ A

For x = 10 ∈ A we have y = - 3/2 ∉ A

For x = 12 ∈ A we have y = - 3 ∉ A

Thus the relation as a set ordered pairs is given by

R = {(0,6) , (4,3) , (8,0)}

Step 3 of 4 :

Find the domain of the relation

The domain of the relation R

= { x ∈ A : (x,y) ∈ R }

= {0, 4, 8}

Step 4 of 4 :

Find the range of the relation

The range of the relation R

= { y ∈ A : (x,y) ∈ R }

= {0, 3, 6}

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Answer 2

In domain and range of a relation, if R be a relation from set A to set B, then

• The set of all first components of the ordered pairs belonging to R is called the domain of R.

Thus, Dom(R) = {a ∈ A: (a, b) ∈ R for some b ∈ B}.

• The set of all second components of the ordered pairs belonging to R is called the range of R.

Thus, range of R = {b ∈ B: (a, b) ∈R for some a ∈ A}.

Therefore, Domain (R) = {a : (a, b) ∈ R} and Range (R) = {b : (a, b) ∈ R}

Note:

The domain of a relation from A to B is a subset of A.  

The range of a relation from A to B is a subset of B.