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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 $R=\{x\in A|\mathit{xRy}$, for some \(y ∈ B\}\)

The co-domain of the relation \(R\) is \(B\)

The range of the relation $R=\{y\in A|\mathit{xRy}$, 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.

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:

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.

**i)**What is the Domain, Co-domain and Range of \(R\).

**ii)**Obtain the Cartesian product.

**iii)**Write the relation in Roster form.

**Solution**:

**i)**The Domain of \(R =\) \(\{a, b, c, d\}\).

Co-domain of \(R =\) \(\{1, 2, 3\}\).

Range of \(R =\) \(\{1, 2\}\).

**ii) The Cartesian product**:

\(A × B =\) \(\{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3), (d, 1), (d, 2), (d, 3)\}\).

**iii) Relation in Roster form**:

A roster form is a method of listing all the elements of a set inside a bracket.

Therefore, the roster form of \(R =\) \(\{(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2), (d, 1), (d, 2)\}\).