UPSKILL MATH PLUS

Learn Mathematics through our AI based learning portal with the support of our Academic Experts!

Learn more
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={xA|xRy, for some \(y ∈ B\}\)
 
The co-domain of the relation \(R\) is \(B\)
 
The range of the relation R={yA|xRy, for some \(y ∈ A\}\)
 
From these definitions, we note that domain of RA, co-domain of \(R = B\) and range of RB.
 
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.
 
1 Ресурс 1.svg
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.
 
2 Ресурс 1.svg
 
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 \(R ∈ A\)
 
3 Ресурс 1.svg
Co-domain:
A co-domain is a set that includes all the possible values of a given function.
The co-domain of the relation \(R = ∈ B\)
 
4 Ресурс 1.svg
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 \(R\).
 
5 Ресурс 1.svg
 
If the teacher cancels the Science examination, then the domain and co-domain will be the same, but the range will be like:
 
6 Ресурс 1.svg
 
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.
 
7 Ресурс 1.svg
 
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 \(R =\) \(\{(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2), (d, 1), (d, 2)\}\).