UPSKILL MATH PLUS

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

The co-domain of the relation $$R$$ is $$B$$

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

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 $$R =$$ $$\{(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2), (d, 1), (d, 2)\}$$.