Domain of a relation

Before understanding the Domain and Range of a Relation, we must understand what a Relation means. A relation is a rule that connects elements in one set to those in another. If A and B are non-empty sets, then the relationship would be defined as a subset of Cartesian Product AxB. This article will tell you about how to find the domain and range of relations. 

How To Find domain and range of relation?

The domain is the set of all ordered pairs’ initial values. The range, on the other hand, is the collection of all ordered pairs of second elements. It also comprises the elements that are used by the function. There is a tricky part in the range, in that Set B can be equal to or larger than the range of the relation. It is because Set B may contain components that are unrelated to Set A.

Domain and range definition 

The Domain of R is the set of initial items belonging to the ordered pair if R is a relation from Set X to Set Y. Mathematical equation for representing domain: 

Dom(R) = {c∈ X: (c, d) ∈R for some d ∈Y  }

The Range of R refers to the set of Second Components that belong to the ordered pair. It can be written as follows:

Range(R) = {d ∈Y: (c, d) ∈R for certain c∈ X}

Domain (R) = {c: (c, d) ∈R }and Range (R) ={ d: (c, d)∈ R} denote Domain and Range, respectively.

Let’s understand it more clearly with the help of examples

Question 1: Find domain and range of the following relation: which has eye color, student’s name, and asset values. Determine whether this relation is a function.

A = {(black, Anne), (brown, Arthur), (green, August), (brown, George), (blue, James), (black, Jonathan)}. Prove whether the given relation is a function or not?

Answer: Domain: {blue, green, brown,black} Range: {Anne, Arthur, August, George, James, Jonathan}

No, the relation is not a function because the set values eye has repeated or wrong colors on it.  

Question 2: If A = {2, 4, 6, 8)   B = {5, 7, 1, 9}.

Assume A relates to B if for a∈A and b∈B a

Answer: Under this relation (R), we have 

R = {(4, 5); (4, 7); (4, 9); (6, 7); (6, 9), (8, 9) (2, 5) (2, 7) (2, 9)}

Therefore, Domain (R) = {2, 4, 6, 8} and Range (R) = {1, 5, 7, 9}

Question 3: In the given ordered pair (5, 7); (9, 5); (5, 5); (10, 12); (7, 4); (4, 1); (3, 4) determine the following relations. Also, determine the domain and range of the given set. 

(a) A R B if for a∈A and b∈B b>a-1

(b) A R B if for a∈A and b∈B a>b-3

(c)  A R B if for a∈A and b∈B a-3>b

(d) A R B if for a∈A and b∈B a=b

Answer:

1. Let R₁ be the set of all ordered pairs in which the first component is less than the second component by 2. Therefore, R₁ = {(5, 7); (10, 12)}

Also, Domain (R₁) = all the x component of R₁ = {5, 10} and Range (R1) = all the y component  of R₂ = {7, 12}

1. R₂ is the set of all ordered pairs in which the first component is less than the second component by a difference greater than equal to one.

Therefore, R₂ = {(5, 7); (10, 12); (3, 4)}. Also, Domain (R₂) = {5, 10, 3} and Range (R₂) = {7, 12, 4}

2. R₃ is the set of all ordered pairs in which the first component is greater than the second component by a difference greater than equal to one.

Therefore, R₃ = {(9, 5); (7, 4); (4, 1)} Also, Domain (R₃) = {9, 7, 4} and Range (R₃) = {5, 4, 1}

3. R₄ is the set of all ordered pairs in which the first and second component are equal

Therefore, R₄ = {(5, 5)}

Also, Domain (R) = {5} and Range (R) = {5}

Question 4: Let A = {3, 4, 5, 6} and B = {9, 10, 11, 12}. A R B if for a∈A and b∈B a is a factor of b. Then write R in the roster form. Also, determine Domain and Range of R.

Answer: Clearly, R consists of elements (a, b) where a is a factor of b. Therefore, Relation (R) in the roster form is R = {(3, 9); (3, 12); (4, 12); (5, 10), (6, 12)}

Domain (R) =all x components of R = {3, 4, 5, 6} and Range (R) = all x components of R= {9, 10, 12}

Question 5: Let A = {2, 3, 4, 5, 6} and B = {p, q, r, s}. Let R be a relation from A in B defined by

R = {2, p}, (2, r), (4, p), (5, q), (6, s), (4, p)}. Determine the domain and range of R.

Answer: Given R = {(2, p), (2, r),(4, p), (5, q), (6, s)} 

The main domain of R = all x components of R = {2,4,5,6}

Range of R = set of second components of all elements of R = {p, r, q, s}

Question 6: If there is a given function f = 2x+1, Find the domain range if 1≤x≤5.

Answer: The values of x lie from 1 to 5.  So, the domain is {1, 2, 3, 4, 5}.  The values of f obtained by putting domain elements in the function will be range.  So, if x=1 then f = 2(1)+1 = 3,

Then x=2 then f = 2(2)+1 = 5, 

if x=3 then f = 2(3)+1 =7, 

if x=4 then f = 2(4)+1 = 9, 

if x=5 then f = 2(5)+1 = 11. So, the range is {3,5,7,9,11}.

Question 7: In this given function:- F = x2 -5, find the domain?

Answer: The value of x is squared and then 5 is subtracted by 5. Any real number can be squared and 5 should be subtracted from it, so there are no restrictions on the domain of this function. The domain is the set of real numbers. The interval form of the domain set will be (-∞,∞).

Conclusion 

A Relation is a collection of predetermined sets or a combination of sets. A point is an organized pair of objects. A relation is a set of input and output values that are displayed as an ordered pair. This article tells you about how to find domain and range, the domain of a relation, and the domain and range of a function.