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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 $\subseteq $ 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}

Please post single query in a thread.

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