Arrow diagram: Domain, Co-domain, and Range — lesson. Mathematics State Board, Class 10.

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Relation:

Let us take any two non-empty sets as

$A$ and

$B$ . A ‘relation’

$R$ from

$A$ to

$B$ is a subset of

$A×B$ satisfying some specified conditions. If

$x ∈ A$ is related to

$y ∈ B$ through

$R$ , then we write it as

$x Ry$ .

$x Ry$ if and only if

$(x,y) ∈ R$ .

Here, the domain of the relation

$R = {x \in A | xRy$ , for some

$y ∈ B\}$

The co-domain of the relation

$R$ is

$B$

The range of the relation

$R = {y \in A | xRy$ , for some

$y ∈ A\}$

From these definitions, we note that domain of

$R \subseteq A$ , co-domain of

$R = B$ and range of

$R \subseteq B$ .

Let us learn in detail about domains, co-domains and ranges using an arrow diagram. Now you may raise a question like, what do you mean by an arrow diagram? Then the answer will be:

An arrow diagram gives a visual representation of the relations.

Let us take the situation which we saw earlier to understand the arrow diagram.

Students $A$	Examinations $B$
Kavya $k$	English $e$
Vimal $v$	Mathematics $m$
Raju $r$	Science $s$
Nancy $n$

Using the above details, we draw the arrow diagram as follows.

The Cartesian product:

$A × B =$

$\{(k, e), (v, e), (r, e), (n, e), (k, m), (v, m), (r, m), (n, m), (k, s), (v, s), (r, s), (n, s)\}$ .

Mark the respective students with their examinations.

Now we learn about the arrow diagram. Let's explore some more concepts using the same scenario.

Domain:

The domain is the set used as an input in a function.

In the above figure, a set

$A$ is said to be the domain of the function.

The domain of the relation

$R ∈ A$

Co-domain:

A co-domain is a set that includes all the possible values of a given function.

The co-domain of the relation

$R = ∈ B$

Range:

The range is the set of values that actually do come out. Range is the co-domain's subset. Remember, all ranges are co-domains but not all the co-domains are ranges.

The range of this scenario is:

Range of

$R = \{e, m, s\}$ , which is

$=$ Co-domain of

$R$ .

If the teacher cancels the Science examination, then the domain and co-domain will be the same, but the range will be like:

Observe the above figure, and tell me how the range differs from the co-domain?

We can notice that the co-domain is the set of all the possible values of a function. However, the range is the set of values that actually do come out.

So, now we understand how the range differs from the co-domain. Let's see an example where we apply all the concepts we discussed.

Example:

Observe the below arrow diagram, which shows the relation between the sets

$A$ and

$B$ , and answer the questions.