Relations and Functions
Ordered Pair
An ordered
pair (x, y) is a pair of objects in which x is called the first component and y
is called the second component. For example, if there are two numbers 4 and 5,
then (4, 5) forms one ordered pair and (5, 4) forms another ordered pair.
Properties of Ordered Pair
1. Order of
the components is very important in an ordered pair. For any numbers a and b,
(a, b) is not equal to (b, a). For example, (4, 7) is not equal to (7, 4).
2. Both the
components of an ordered pair can be same. For any number a, (a, a) can be the
ordered pair. For example, (7, 7), (9, 9), etc.
3. Two
ordered pairs are equal if the first component of one ordered pair is equal to
the first component of the second ordered pair. Same is true for the second
component. For example, (11, 2) = (11, 2) but (11, 2) ≠ (11, 3).
Cartesian Product
The
cartesian product of two non-empty sets P and Q is the set of all ordered pairs
whose first component is an element of set A and the second component is an
element of set B. A cartesian product is shown as P × Q and is read as ‘P cross
Q’.
For example,
if P = {2, 4, 6} and Q = {5, 10, 15, 20} then P × Q = {(2, 5), (2, 10), (2,
15), (2, 20), (4, 5), (4, 10), (4, 15), (4, 20), (6, 5), (6, 10), (6, 15), (6,
20)}
Note that
1. P × Q ≠ Q × P when P ≠ Q. For example, if P = {2, 3}, Q =
{5, 6}, then P × Q = {(2, 5), (2, 6), (3, 5), (3, 6)} but Q × P = {(5, 2), (5,
3), (6, 2), (6, 3)}.
2.
Conversely, if P = Q, then P × Q = Q × P. For example, if P = {4, 5} and Q =
{4, 5}, then P × Q = {(4, 4), (4, 5), (5, 4), (5, 5)} and Q × P = {(4, 4), (4,
5), (5, 4), (5, 5)}.
Thus, P × Q
= Q × P.
3. Number of
ordered pair is same in P × Q and Q × P, i.e., n (P × Q) = n (Q × P). For example, if P = {8, 10} and Q
= {9, 11}, then n (P × Q) = 4 and n (Q × P) = 4. Thus, n (P × Q) = n (Q × P).
Relation
We can see
some kind of a connection or association between two objects in our daily life.
For example,
a. Anu is
elder than Sam. It shows the connection between two persons.
b. 1 is
greater than –2. It shows the connection between two numbers.
Thus, the
connection or association between two objects is defined as relation.
Mathematically, any set of ordered pair is called a relation. Relation is
generally represented using the letter R. Therefore in the first example,
relation R = ‘is elder than’ and in the second example, relation R = ‘is
greater than’.
Let A and B
be two non-empty sets. Then the relation between an element a of set A and an
element b of set B is written as a R b. For example, let P = {0, 2, 3, 4}, Q =
{9, 16} and R be the relation defined from P to Q as ‘is a factor of’, then we
get 3 R 9, 2 R 16, 4 R 16.
Representation of a Relation
There are
three methods of representing relation.
1. Roster
form
2. Set
builder form
3. Arrow
diagram
Roster Form
In this type
of representation, a relation R from set A to set B is shown by the set of all
ordered pairs (a, b) such that a ϵ A and b ϵ B which satisfies the given relation R.
Let A = {3,
7, 9, 15}, B = {2, 8} and R be the relation ‘is more than’ from A to B, then we
get R = {(3, 2), (7, 2), (9, 2), (15, 2), (9, 8), (15, 8)}
Set Builder Form or Equation Form
In this
method, the relation R from A and B is represented using statements or formulae
in which an element a ϵ A and an element b ϵ B.
Let A and B
be two non-empty sets, then the relation R ‘is a square of’ from A to B is
shown as R = {(a, b) : a ϵ A, b ϵ B and a is a square of b}.
Arrow Diagram
In this
method, the relation R from A to B is represented using arrows from set A to
set B satisfying the given relation.
Let A and B
be two non-empty sets, then the relation R ‘is a cube root of’ from A to B
where A = {2, 3, 4, 5} and B = {8, 27, 64, 125} is represented as shown.
Domain and Range of a Relation
The set of
first components of the ordered pairs in a relation is its domain and the set
of second components of the ordered pairs in a relation is its range.
Let X = {1,
2, 3, 4, 5, 6, 7}, Y = {1, 4, 9, 16, 18} and the relation R ‘is a square root
of’ from X to Y, then R = {(1, 1), (2, 4), (3, 9), (4, 16)}, domain = {1, 2, 3,
4} and range = {1, 4, 9, 16}.
Example
1: Let P = {1, 2, 3,
4, 5, 6}, Q = {16, 24, 33} and the relation R ‘is a prime factor of’ from P to
Q. Then
a. represent
R in roster form, set builder form and using arrow diagram.
b. find the
domain and range of R.
Solution: Given, P = {1, 2, 3, 4, 5, 6}, Q =
{16, 24, 33} and relation R ‘is a prime factor of’ from P to Q.
Roster form:
R = {(2, 16), (2, 24), (3, 24), (3, 33)}
Set builder
form: R = {(p, q): p ϵ P, q ϵ Q and p is a prime factor of q}
Arrow
diagram:
b. Domain of
R = {2, 3}, Range of R = {16, 24, 33}
Function or Mapping
A function
or mapping from A to B is a special type of relation where A, B be two
non-empty sets in which every element a of set A is related to a unique element
b of set B. A function is written as f: A → B, i.e., f is a function from A to B
and it means that f is a function or mapping from A to B and is denoted by f
(x). Also, set A is called the domain of the function and set B is the range of
the function which is also known as co-domain.
Conditions for Functions or Mapping
1. Every
element of set A is associated with one element of set B, i.e., there should
not be any element of set A which is not associated with any element of set B and
no element of set A should be associated with 2 or more elements of set B.
2. If f is a
function from A to B (f : A → B), then no two ordered pairs should
have the same first component.
Representation of a Function
Let A and B
be two non-empty sets and f : A → B is represented as:
Roster Form
Let A = {1,
8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A → B have relation ‘is cube of’ is represented in roster form
as f = {(1, 1), (8, 2), (27, 3), (64, 4)}.
Equation Form
Let A = {1,
8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A → B have relation ‘is cube of’ is represented in equation form
as f (x) = y = 3√x, where x ϵ A and y ϵ B.
Set Builder Form
Let A = {1,
8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A → B have relation ‘is cube of’ is represented in set builder
form as f = {(x, y): x ϵ A, y ϵ B and y = 3√x}.
Arrow Diagram
Let A = {1,
8, 27, 64}, B = {1, 2, 3, 4, 5} and function f : A → B have relation ‘is cube of’ is represented in arrow diagram
as:
Value of the Function
Let A and B
be two non-empty sets, then f(x) is known as the value of the function at x. If
f : x → 3x – 4, then f(x) = 3x – 4
Thus, the
value of the function at x = 3 is f(3) = 3 × 3 – 4 = 5.
Example
2: State whether the
relations shown below are functions or not.
a. If A =
{a, b, c, d} and B = {x, y, z}, then R = {(a, x), (b, y), (c, x), (d, y)}
b. If A =
{1, 2, 3, 4, 5} and B = {6, 7}, then R = {(1, 6), (2, 7), (3, 7), (4, 7)}
c. If A =
{10, 20, 30, 40, 50} and B = {a, b, c}, then R = {(10, b), (20, b), (30, b),
(40, b), (50, c)}
Solution:
a. The
relation R is a function as every element of set A is related to a unique
element of set B.
b. The
relation R is not a function as 5, an element of A is not related to any
element of set B.
c. The
relation R is a function as every element of set A is related to an element of
set B.