Question 1: If ,
, which of the following are relations from
to
? Give reasons in support of your answer: (i)
(ii)
(iii)
{iv}
Answer:
Given ,
i) is not a relation from
to
as it is not a subset of
ii) is a subset of
. So it is a relation from
to
.
iii) is not a relation from
to
as it is not a subset of
iv) is a relation from
to
.
Question 2: A relation is defined from a set
to a set
as follows:
is relatively prime to
. Express
as a set of ordered pairs and determine its domain and range.
Answer:
Given and
Also is relatively prime to
Therefore is a co-prime to
and
is a co-prime to
and
is a co-prime to
and
is a co-prime to
and
Hence Domain of
and Range of
Question 3: Let be the set of first five natural numbers and let
be a relation on
defined as follows:
. Express
and
as sets of ordered pairs. Determine also (i) the domain of
(ii) the range of
.
Answer:
Given
is the set of first five natural numbers
Also it is given that R is a relation on A defined as
i) Domain
ii) Range
Question 4: Find the inverse relation in each of the following cases: (i)
(ii)
(iii)
is a relation from
to
defined by
.
Answer:
i) Given
ii) Given
Now
Putting we get
respectively
For , we get
and for
Hence
iii) Given is a relation from
to
defined by
Now,
Putting we get
respectively.
and
Question 5: Write the following relations as the sets of ordered pairs:
(i) A relation from the set
to the set
defined by
.
(ii) A relation on the set
defined by
is relatively prime to
.
(iii) A relation on the set
defined by
.
(iv) A relation from a set
to the set
defined by
divides
.
Answer:
i) Given ,
Putting we get
respectively
ii) Given relation on the set
defined by
is relatively prime to
Therefore is relatively prime to
and
is relatively prime to
and
is relatively prime to
and
is relatively prime to
and
is relatively prime to
and
iii) Given
Putting we get
respectively
iv) Given a relation from a set
to the set
defined by
divides
Here divides
and
divides
and
divides
Question 6: Let be a relation in
defined by
. Express
and
assets of ordered pairs.
Answer:
Given
Now
Putting we get
respectively
For , we get
and for
Hence
Question 7: Let and
. Let
. Show that
is an empty relation from
into
.
Answer:
Given and
Also
For elements of and
They are all even
Therefore R is an empty relation from to
.
Question 8: Let and
. Find the total number of relations from
into
.
Answer:
Given and
Therefore there are relations from
to
total number of relations
Question 9: Determine the domain and range of the relation R defined by:
(i)
(ii)
Answer:
i) Given
Therefore for elements given we get
Therefore Domain
Range
ii) Given
Therefore Domain
Range
Question 10: Determine the domain and range of the following relations
(i)
ii)
Answer:
i) Given
Therefore Domain
Range
ii) Given
respectively
Therefore Domain
Range
Question 11: Let . List all relations on
and find their number.
Answer:
Given
We know that number of relations
Ordered pairs
Question 12: Let and
. Find the total number of relations from
into
.
Answer:
Given and
and
Therefore there are relations from
to
Question 13: Let be a relation from
to
defined by
and
. Are the following statements true?
(i) for all
(ii)
(iii) and
Answer:
Given and
i) for all
False: Statement is not true because
ii)
False: The statement is not true because but
iii) and
False: Statement is not true because and
but
Question 14: Let . Define a relation on a set
by
. Depict this relationship using an arrow diagram. Write down its domain, co-domain and range.
Answer:
Given
Putting then
respectively
For which does not belong to
Therefore Domain
Range
Question 15: Define a relation on the set
of natural numbers by
Depict this relationship using (i) roster form (ii) an arrow diagram. Write down the domain and range or R.
Answer:
Given
i) Putting we get
respectively.
ii) Domain
Range
Question 16: and
. Define a relation
from
to
by
. Write
in Roster form.
Answer:
Given and
Also
For
For
For
For
Question 17: Write the relation in roster form.
Answer:
Given
Hence
Question 18: Let . Let
be a relation on
defined by
(i) Write
in roster form (ii) Find the domain of
(iii) Find the range of
Answer:
Given and
be a relation on
defined by
For
For
For
For
For
For
i)
ii) Domain
iii) Range
Question 19: Figure below shows a relationship between the sets
and
. Write this relation in (i) set builder form (ii) roster form. What is its domain and range?
Answer:
Given
and
When
Therefore Domain
and Range
Question 20: Let be the relation on
defined by
. Find the domain and range of
.
Answer:
Given
Difference between two integers would also be an integer.
Therefore Domain
And Range
Question 21: For the relation defined on
by the rule
. Prove that
and
is not true for all
.
Answer:
To prove: and
is not true for all
.
Given
Let
Here,
And,
But,
and
is not true for all
Hence Proved.
Question 22: Let be a relation on
defined by
for all
. Show that:
(i) for all
(ii) for all
(iii) and
for all
Answer:
Given for all
i) for all
for all
for all
ii) for all
iii) and
for all
and
and
Question 23: Find the linear relation between the components of the ordered pairs of the relation , where (i)
(ii)
Answer:
i) Given
and
are related to each other by linear equation
Let the relation be defined as
For we get
For we get
Therefore or
ii) Given
and
are related to each other by linear equation
Let the relation be defined as
For we get
For we get
Therefore
Question 24: If and
, write the relation
as a set of ordered pairs, if
(i)
(ii)
Answer:
Given and
i)
If is even then
ii) if is odd, then