Domain and range of a relation

Domain and range of a relation

The domain of a relation

Domain: The domain of a function is the set of all possible input values (often the "x" variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathematically defined.

The range of a relation

Range: The range is the set of all possible output values (usually the variable y, or sometimes expressed as f(x)), which result from using a particular function.

If R is the relation on two sets A and B such that set A is an independent set while B is the dependent set, then set A is the Domain while B is the Co-domain or Range.

Note that each member of set A must be mapped to at least one element of set B and each member of set B must be an image of at least one element in set A.

Consider the following relation

For the above relation, the domain is {1, 3} while the range is {a, b}

Example

Let P = {1, 3, 4, 10} and Q = {0, 4, 8}

Find the domain and range of the relation R: "is less than"

For the pictorial representation of the relation R above the Domain is {1, 3, 4} and the Range is {4, 8}

Example

Let

$R = \left\{ (x, y) : y = x + 1 \quad \text{and} \quad -2 < x \leq 8 \right\}$

Where $R$ is the relation and both $x$ and $y$ are integers.

State the domain and range of $R$:

Solution

Domain

$\text{Domain} = \left\{ x : -2 < x \leq 8 \right\} = \{-1, 0, 1, 2, 3, 4, 5, 6, 7, 8\}$

Range

$\text{Range} = \left\{ y : y = x + 1 \right\} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$

$\therefore \text{Domain} = \{-1, 0, 1, 2, 3, 4, 5, 6, 7, 8\} \quad \text{and} \quad \text{Range} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$