| Relations,
Functions and
Graphs |
| Functions |
| x = {1, 2, 3, 4, 5} and y = {2, 4, 6, 8, 10}
A
function, f, from
x to y is a relation
between x and y such that for each element
in set x there
is one, and only one, that is associated to an element
in set y. The
set x is called
the domain of
the function,
y is called its range.
The
function consists
of two parts:
1.
The rule - which tells you how the values
are to be calculated.
2.
The domain - this
tells you the set
of values to which
the rule can be applied |
| Relations,
Ordered Pairs, Domain and Range |
A relation is a set of ordered pairs (x, y),
where x is an element of set x and y
is an element of set y.
In
the ordered pair (x, y),
the domain of
a function is
the set of all
“first coordinates” of the ordered pairs of a relation and the range of a function is the set of
all “second coordinates” of the ordered pairs of a relation.
Therefore,
x would be considered to be the first coordinate and
y, the second coordinate.
For
example, if
x
= {1, 2, 3, 4, 5} and y = {2, 4, 6, 8, 10}
Then
the function,
f, is x × 2 or
f(x)
= 2x
The ordered pairs would be {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}
The domain would
be {1, 2, 3, 4, 5} and the range would be {2, 4, 6, 8, 10}
f(x) = 2x
x
------------------> y
1
------------------> 2
2 ------------------>
4
3 ------------------>
6
4 ------------------>
8
5 ------------------> 10
Another
word for function is mapping. A function is said
to map an element in its domain to an element in its range.
In
this example, we can say that 1 maps to 2, 2 maps to
4, 3 maps to 6
and so on.
Here,
it is a one-to-one mapping since one element of set x maps to only one element in set y. |
| Graph
Diagrams |
A relation can also
be represented through the use of a graph.
|
{(1,
1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1),
(3, 2), (3, 3)}
are
the ordered pairs represented as blue dots on the
graph to the left. |
|
| Arrow
Diagram |
Through
the use of an arrow diagram a relation can also be represented.
 |
Set
A is a Capital of Set B |
|
{(London,
Great Britain), (Paris, France), (New Delhi, India),
(Washington, United States)} |
|
| Types
of Functions |
The
types of mappings are one-to-one, one-to-many, many-to-one and many-to-many.
 |
one-to-one |
|
A mapping that is one-to-one exists
when one element of the domain maps to only one element of the range. |
|
A mapping that is many-to-one exists
when many elements of the domain map to one element of
the range. |
| |
A mapping that is one-to-many exists
when one element of the domain maps to many elements of the range. |
| |
A mapping that is many-to-many exists
when many elements of the domain map to many elements of the range. |
|
 |
|
|
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