Download Examples of relations

RELATIONS
Plan
•Definition with various examples
based on our daily life.
•Presentation with diagrams.
•Presentation with examples.
•Idea of its different types.
•Focusing on solution of
assignments.
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Ordered Pairs
Let Mrs. Rama has three suits A= {S1, S2, S3}
and two necklaces B= {N1, N2}. She is going to
attend a reception party with her husband
Ramesh. She has to find the best match of suits
and necklaces i.e. she has to make pair of suits
and necklaces in the following manner:
(S1,N1),(S1,N2),(S2,N1),(S2,N2),(S3,N1)
and
(S3,N2)
Thus 6 distinct ordered pairs can be
constructed.
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•Discussion of previous knowledge required to
define a relation.
Ordered Pairs: Let us consider the two sets say A
and B such as A= {a, b} and B={p, q}. Let us form
pairs of elements taking one from each set. Then
pairs are (a , p) ,(a, q),(b, p),(b, q). We call such
pairs as ordered pairs. Thus we can define an
ordered pair as follows:
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Ordered pairs
An ordered pair is a pair of elements in
specified order separated by a comma and
enclosed within parentheses (small brackets).
For example
(a) (a,2), (b,3), (c,4), (d,5) are ordered pairs
whose first components a, b, c and d are English
alphabets taken in order and second
components 2,3,4 and 5 are natural numbers.
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(b)
Points in a plane are also represented by
ordered pairs (x, y) for abscissa’s ‘x’ and ordinates
‘y’.
Equality of Ordered Pairs: Two ordered pairs (x, y)
and (m, n) are said equal if x=m and y=n.
Cartesian Product of two sets: Let us
consider two sets A={Ram, Shyam}, B={Sita,
Radha}. Then possible ordered pairs are (Ram,
Sita),
(Ram, Radha), (Shyam, Sita), (Shyam,
Radha)on
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considering first components from set A and
second components from set B. The set of all
these ordered pairs is called the Cartesian
product of set A and B and is written as
AXB={(Ram, Sita), (Ram, Radha), (Shyam, Sita),
(Shyam, Radha)}.
Thus we can define Cartesian product as follows:
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Definition
The Cartesian product of two non-empty sets is
the set of all ordered pairs with first components
from first set and second components from
second set. Symbolically the Cartesian product of
A and B is
If either of A and B is the null set then A X B is
null set.
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Cardinal number of
Cartesian product of sets
Let A and B be finite sets then the number of
all the ordered pairs of A X B is called Cardinal
number of A X B . We write Cardinal of a set A as
n (A).
n (A X B) = n(A).n(B)
And n(B X A) = n(B).n(A)
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For example:
A = {1, 2}, B = {a, b} then
A X B = {(1, a), (1, b), (2, a), (2, b)}
n (A X B) = 4 and n(A)=2, n(B)=2
n (A X B) = 4 and n(A)=2,
n(B)=2
n (A X B) = n(A) . n (B)
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Relation:
Relation is familiar in our daily life such as husband and
wife, father and son, brother and sister, mother and
daughter etc. In Mathematics we observe ‘a line l is
perpendicular to line m’, ‘a line l is parallel to line m’,
∆PQR is similar to ∆ XYZ’, ‘∆ ABC is congruent to
∆LMN’, ‘‘ Set A is equal to Set B’, ‘Set A is subset of Set B’
etc. In these relations we see that a pair of objects is
involved in specific order. Let us consider two sets and
introduce a relation between the objects from first set to
second set. For example: (1) A={Ram, Bharat} ,
B={Dashrath } . There is a relation ‘ is son of ‘ between the
elements of set A and set B. If we denote R for the
relation ‘ is son of’ then we have Ram R Dashrath,
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Example-2 : Let us consider two sets A={Patna,
Kolkatta, Ranchi}, B={Bihar, West Bengal,
Jharkhand, Madhya Pradesh}. We define a relation
‘is capital of’ then the relation can be represented
as follows:
Figure- 1
A
B
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R = {(2,4), (2,6),(2,10),(3,6),(3,9)}
Obviously
A X B, . With the help of examples we define a
relation as follows: Relation R from
Definition: Let A and B be any two non empty
sets. Then relation R from set A to set B is a
subset of A × B.
Thus a relation is a set of ordered pairs.
Of course, this means that every graph is
relation.
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Examples of relations:
Example : (1) R= {(2,1), (1,3), (3,2),(4,3)} with
corresponding graph in figure
Example : (2) R= {(0,0), (1,1), (2,2),(3,3), (4,4)}
with corresponding graph in figure
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Example : (2) R= {(0,0), (1,1), (2,2),(3,3), (4,4)}
with corresponding graph in figure
Figure - 4
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Binary Relation on a set :
If A be a non-empty set, then a subset of A X A is
called a binary relation or simply a relation on A.
Example: Let N be the set of natural numbers.
There is a relation ‘ has as its square’ from the set
N to N. If we write R for the relation ‘has as its
square’, then 1R1, 2R4, 3R9, 4R16, 5R25,
…………………
In the form of ordered pairs we write R = {(1,1),
(2,4), (3,9), (4,16), (5,25), ………..}
In a diagram we represent it as
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N
N
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Domain, Co - domain
and Range of a Relation :
Let R be a relation from set A to set B , then the
domain of the relation R, denoted by D, is the set of
all first components of the ordered pairs of R
The range of R the set of all second components
of the ordered pairs of R
i.e. E = {y: y
}
If R is a relation from set A to set B then set B is
called co-domain of R.
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Example: (1)
Let A = {Delhi, Kabul, Kathmandu}
B = {India, Sri Lanka, Afghanistan, Nepal, Pakistan}
We define a relation R as ‘is capital of’, then
R = { (Delhi, India), (Kabul, Afghanistan),
(Katmandu, Nepal)}.
Then domain of R is
D = {Delhi, Kabul, Katmandu}
Range E of R is
E = {India, Afghanistan, Nepal}
And Co-domain of R is
{India, Sri Lanka, Afganistán, Nepal, Pakistán}
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Example: (2)
Let A = {1, 2, 3 ,4}
B = { 1,2, 3, }
R is defined with the graph as follows:
i.e, R = { (1, 3), (2, 1), (3, 2) , (4,3)}
Then D = { 1, 2, 3,4} is domain of R
E = {1, 2, 3} is range of R
And set B = { 1 , 2, 3 } is co-domain of R
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Example (3): Let us consider a relation R between
the sets X and Y follows:
The relation R can be written in roster form as
R = {(5, 3), (6, 4), (7, 5)}
In set builder form, the relation R can be
expressed as
Then domain ‘D’ of R is, D = { 5, 6, 7 }
Range E of R is, E = {3, 4, 5 }
And set Y = { 3, 4, 5 } is co-domain of R.
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Figure – 8
From this family tree we see that Ramesh is son
of Rakesh and Reshma . Rimjhim is daughter of
Radha and Ramesh. Let us define a relation R as
the set of ordered pairs (x, y) where x = a
daughter and y = the mother of x then the
relation is
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R = { (Kajal, Reshma), (Rimjhim, Radha),
(Kritika, Kajal) }
Then domain of R is
D = {Kajal, Rimjhim, Kritika }
Range of R is
E = { Reshma , Radha, Kajal }.
If we define a relation R , where x = a
brother and y = the sister of x. Then this
relation is R = {(Ramesh, Kajal ),(Ravi,
Rimjhim) }
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Thus domain D of R is
D = { Ramesh , Ravi } and the range E of R is
E = { Kajal, Rimjhim }
In this way many examples can be produced for
Domain, Co-domain and Range of a relation R.
Points to remember
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Some Particular types of Relations
Empty relation: If A be a non-empty set
called void relation or empty relation.
i.e. no element of A is related to any element of A.
R is an empty relation in A.
Universal Relation: A relation R in a set A is called
universal relation, if each element of A is related to
every element of A.
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Example : Let T be the set of all triangles
in a plane. A relation R on T defined by
R = { (T1, T2) : T1 is congruent to T2} , R is
reflexive as every triangle is congruent to
itself.
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Example (2): The identity relation on
a non-empty set A is symmetric.
Example (3) : The universal relation
on a non-empty set A is symmetric.
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