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