Meaning of definition 2,3,4
Definition 2 A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product AxB. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A X B. The second element is called the image of the first element.
Definition 3 The set of all first elements of the ordered pairs in a relation R from a set A to set B is called the domain of the relation R.
Definition 4 The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R. Note that range codomain.
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.
For Example:
If A = {2, 4, 6, 8) B = {5, 7, 1, 9}.
Let R be the relation ‘is less than’ from A to B. Find Domain (R) and Range (R).
Solution:
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}
Example-2If there is a relation,
R₂ = {(4, 6); (9, 11); (2, 3)}.
Also, Domain (R₂) = {4, 9, 2} and Range (R₂) = {6, 11, 3}
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