The concept of relation in math refers to an association of two objects or two variables based
some property possessed by them.
For Example:
1. Rachel is the daughter of Noah.
This statement shows the relation between two persons.
The relation (R) being ‘is daughter of’.
2. 5 is less than 9.
This statement shows the relation between two numbers.
The relation (R) being ‘is less than’.
If A and B are two nonempty sets, then the relation R from A to B is a subset of A x B, i.e., R ⊆ A x B.
If (a, b) ∈ R, then we write a R b and is read as 'a' related to 'b'.
3. Let A and B denote the set animals and their young ones.
Clearly, A = {cat, dog, cow, goat}
B = {kitten, puppy, calf, kid}
The relation (R) being ‘is young one of ‘.
Then the fact that,
Kitten is the young one of a cat.
Thus, kitten is related to cat.
Puppy is the young one of a dog.
Thus, puppy is related to dog.
Calf is the young one of a cow.
Thus, calf is related to cow.
Kid is the young one of a goat.
Thus, kid is related to goat.
This fact can also be written as set R or ordered pairs.
R = {(kitten, cat), (puppy, dog), (calf, cow), (kid, goat)}
Clearly, R ⊆ B × A
Thus, if A and B are two nonempty sets, then the relation R from A to B is a subset of A×B, i.e., R ⊆ A × B.
If (a, b) ∈ R, then we write a R b and is read as a is related to b.
The relation in math from set A to set B is expressed in different forms.
(i) Roster form
(ii) Set builder form
(iii) Arrow diagram
i. Roster form:
● In this, the relation (R) from set A to B is represented as a set of ordered pairs.
● In each ordered pair 1st component is from A; 2nd component is from B.
● Keep in mind the relation we are dealing with. (>, < etc.)
For Example:
1. If A = {p, q, r} B = {3, 4, 5}
then R = {(p, 3), (q, 4), (r, 5)}
Hence, R ⊆ A × B
2. Given A = {3, 4, 7, 10} B = {5, 2, 8, 1} then the relation R from A to B is defined as ‘is less than’ and can be represented in the roster form as R = {(3, 5) (3, 8) (4, 5), (4, 8), (7, 8)}
Here, 1ˢᵗ component < 2ⁿᵈ component.
In roster form, the relation is represented by the set of all ordered pairs belonging to R.
If A = {1, 1, 2} and B = {1, 4, 9, 10}
if a R b means a² = b
then, R (in roster form) = {(1, 1), (1, 1), (2, 4)
ii. Set builder form:
In this form, the relation R from set A to set B is represented as R = {(a, b): a ∈ A, b ∈ B, a...b}, the blank space is replaced by the rule which associates a and b.
For Example:
Let A = {2, 4, 5, 6, 8} and B = {4, 6, 8, 9}
Let R = {(2, 4), (4, 6), (6, 8), (8, 10) then R in the set builder form, it can be written as
R = {a, b} : a ∈ A, b ∈ B, a is 2 less than b}
iii. Arrow diagram:
● Draw two circles representing Set A and Set B.
● Write their elements in the corresponding sets, i.e., elements of Set A in circle A and elements of Set B in circle B.
● Draw arrows from A to B which satisfy the relation and indicate the ordered pairs.
For Example:
1. If A = {3, 4, 5} B = {2, 4, 6, 9, 15, 16, 25}, then relation R from A to B is defined as ‘is a positive square root of’ and can be represented by the arrow diagram as shown.
Here R = {(3, 9); (4, 16); (5, 25)}
In this form, the relation R from set A to set B is represented by drawing arrows from 1ˢᵗ component to 2ⁿᵈ components of all ordered pairs which belong to R.
2. If A = {2, 3, 4, 5} and B = {1, 3, 5} and R be the relation 'is less than' from A to B,
then R = {(2, 3), (2, 5), (3, 5), (4, 5)}
● Relations and Mapping
Domain and Range of a Relation
Domain Codomain and Range of Function
● Relations and Mapping  Worksheets
Worksheet on Functions or Mapping
7th Grade Math Problems
8th Grade Math Practice
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