Here we will discuss about the functions and relations. It's very fascinating and confusing topic for everyone who is not a master in mathematics. It's a wide topic which needs a lot of time and effort to get grip on this topic. Let's begin with Relations.
We know that Delhi is the capital of India. London is the capital of UK and Jeddah is the capital of KSA. A set representing the ordered pair depicting these Nations with their capitals may be written as {(India,Delhi), (UK,London), (KSA,Jeddah)}.
Here we can notice that elements of this set are ordered pairs and elements in ordered pairs are according to rule second one is the capital of first one. We can see that second element is the name of the capital of the name of the nation representing by first element.
Here we can notice that elements of this set are ordered pairs and elements in ordered pairs are according to rule second one is the capital of first one. We can see that second element is the name of the capital of the name of the nation representing by first element.
Similarly, we can have another set as {(5,10), (6,12), (7,14), (8,16)}
It's a relation too because second element is associated to first element according to some rule in each ordered pair. We see that in (5, 10), 10 is twice of 5, similarly 16 is twice of 8 in the ordered pair (8,16).
It's a relation too because second element is associated to first element according to some rule in each ordered pair. We see that in (5, 10), 10 is twice of 5, similarly 16 is twice of 8 in the ordered pair (8,16).
We can separate this relation as A={5,6,7,8} and B={10,12,14,16}
And we can write relation $R: A\rightarrow B$
We can also summarize that set A is the domain and set B is the range of this relation.
Thus, domain is the set {5,6,7,8} and range is the set {10,12,14,16}.
This method of writing a relation is known as Roster Method. We can write above relation in set builder form too. R={(x, y): y=2x and x belongs to the set {5, 6, 7, 8}}.
Now we can define a relation as
A relation is the set of the ordered pairs such that elements in ordered pairs are binding some rule.
Now we can define a relation as
A relation is the set of the ordered pairs such that elements in ordered pairs are binding some rule.
and a function is defined as below
A function is defined as the relation in which every element in domain is associated to a unique element in the range.
A function is defined as the relation in which every element in domain is associated to a unique element in the range.
(i) If an element in the domain is not associated to an element in the range of the relation then the given relation is not a function. For example. If A={2,3,4,5} and B={5,6,7} and if x is an element in A and y is an element in B then $R:A\rightarrow B; y=x+2$
We see that 3+2=5, 4+2=6 and 5+2=7
Here 3, 5, 7 are associated to elements in set B but 2 is not associated to any element in B.
Since all the elements are not associated the elements in B, hence, R={(3, 5),(4,6),(5,7)} is not a function from A to B.
We see that 3+2=5, 4+2=6 and 5+2=7
Here 3, 5, 7 are associated to elements in set B but 2 is not associated to any element in B.
Since all the elements are not associated the elements in B, hence, R={(3, 5),(4,6),(5,7)} is not a function from A to B.
It can be depicted graphically as below
Note: Every elements in the domain must be associated to unique elements in the range for a relation to be a function.
(ii) Repetition of the first element in ordered pairs is not allowed. If there is repetition of first element in ordered pairs of a relation then the relation is not a function. Because the term unique is not satisfied for such a relations.
Consider, a relation R={(2, 5), (2, 6), (3, 9)}. Here we see that the first element in ordered pairs (2,5) and (2, 6) is same. Thus there is repetition of first elements. Hence this relation R is not a function too.
Graphically, it can be depicted as
Consider, a relation R={(2, 5), (2, 6), (3, 9)}. Here we see that the first element in ordered pairs (2,5) and (2, 6) is same. Thus there is repetition of first elements. Hence this relation R is not a function too.
Graphically, it can be depicted as
It's not a function too.
Graph of $y^2=2x$ is not a function too as shown below
Graph of $y^2=2x$ is not a function too as shown below
We see that point (2, 2) and (2, -2) both lie on the graph of $y^2=2x$
And hence, $y^2=2x$ does not represent a function.
Similarly any horizontal parabola, ellipse and circle is not a function.
However a linear function is a function if there is no hole in its graph.
Try these problems
Which one is/are function/s in the following:-
And hence, $y^2=2x$ does not represent a function.
Similarly any horizontal parabola, ellipse and circle is not a function.
However a linear function is a function if there is no hole in its graph.
Try these problems
Which one is/are function/s in the following:-
1. R={(7, 8), (8, 9),(8,10), (9, 11)}
2. R={(2, 4),(3, 6),(4, 8)}
3. $y^2=6x$
4. $y=3x+5$
Mohd Naseem Hashmi
5th Oct 2018
Mohd Naseem Hashmi
5th Oct 2018
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