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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 , for some \(y ∈ B\}\)
The co-domain of the relation \(R\) is \(B\)
The range of the relation , for some \(y ∈ A\}\)
From these definitions, we note that domain of , co-domain of \(R = B\) and range of .
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 \(R =\) \(\{(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2), (d, 1), (d, 2)\}\).
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