### Question 12. Let A = {x, y, z} and B = {a, b}. Find the total number of relations from A into B.

**Solution:**

The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A Ã— B.

If n(A) = p and n(B) = q, then n(A Ã— B) = pq.

So, the total number of relations is 2

^{pq}.n(A) = 3 and n(B) = 2 â‡’ n(A Ã— B) = 3 Ã— 2 = 6

âˆ´ Total number of relations = 2

^{6}= 64

### Question 13. Let R be a relation from N to N defined by R= {(a, b): a, b âˆˆ N and a = b^{2}}.

### Are the following statements true?

### (i) (a, a) âˆˆ R for all a âˆˆ N

### (ii) (a, b) âˆˆ R â‡’(b, a) âˆˆ R

### (iii) (a, b) âˆˆ R and (b, c) âˆˆ R â‡’(a, c) âˆˆ R

**Solution:**

Given:R = {(a, b): a, b âˆˆ N and a = b^{2}}.

(i)(a, a) âˆˆ R for all a âˆˆ NHere, take b = 2

a = b

^{2}= 2^{2}= 4âˆ´ (4, 2) R but (2, 2) âˆ‰ R

As, 2

^{2}â‰ 2So,

No, the statement is false.

(ii)(a, b) âˆˆ R â‡’ (b, a) âˆˆ RHere, take b = 2

a = b

^{2}= 2^{2}= 4âˆ´ (4, 2) âˆˆ R but (2, 4) âˆ‰ R

As, 4

^{2}â‰ 2So,

No, the statement is false.

(iii)(a, b) âˆˆ R and (b, c) âˆˆ R â‡’ (a, c) âˆˆ RHere, take b = 4

a = b

^{2}= 4^{2 }= 16(16, 4)âˆˆ R

Now, b = c

^{2}4 = c

^{2}c = -2 âˆ‰ N

(4, 2) âˆˆ R

But (16, 2) âˆ‰ R

As, 2

^{2}â‰ 16So,

No, the statement is false.

### Question 14. Let A= {1, 2, 3,â€¦.,14}. Define a relation on a set A by R= {(x, y): 3x â€“ y = 0, where x, y âˆˆ A}. Depict this relationship using an arrow diagram. Write down its domain, co-domain, and range.

**Solution:**

Given:R = {(x, y): 3x â€“ y = 0, where x, y âˆˆ A}A = {1, 2, 3,â€¦,14}

As, y = 3x

âˆ´ R = {(x, 3x): where x, 3x âˆˆ A}

R = {(1, 3Ã—1), (2, 3Ã—2), (3, 3Ã—3), (4, 3Ã—4)}

Note: We cannot include (5, 3Ã—5) as 15 âˆ‰ A

R = {(1, 3), (2, 6), (3, 9), (4, 12)}

So,

Domain of relation R = {1, 2, 3, 4}

Co-Domain of relation R = {1, 2, 3,â€¦,14} = A

Range of relation R = {3, 6, 9, 12}

### Question 15: Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5, x is a natural less than 4, x, y âˆˆ N}. Depict this relationship using (i) roster form (ii) an arrow diagram. Write down the domain and range of R.

**Solution:**

R = {(x, y): y = x + 5, x is a natural less than 4, x, y âˆˆ N}

Natural numbers less than 4 are 1, 2 and 3.

On putting x = 1, y = 1 + 5 = 6

On putting x = 2, y = 2 + 5 = 7

On putting x = 3, y = 3 + 5 = 8

(i)R = {(1, 6), (2, 7), (3, 8)}

(ii)The arrow diagram represent the relation(R):So,

Domain of relation R = {1, 2, 3}

Range of relation R = {6, 7, 8}

### Question 16: A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd, x âˆˆ A, y âˆˆ B}. Write R in Roster form.

**Solution:**

Given:Relation R from A to B by R = {(x, y): the difference between x and y is odd, x âˆˆ A, y âˆˆ B}

A = {1, 2, 3, 5} and B = {4, 6, 9}

For x = 1:

y â€“ x = 4 â€“ 1 = 3 which is odd â‡’ 4 âˆˆ y

y â€“ x = 6 â€“ 1 = 5 which is odd â‡’ 6 âˆˆ y

y â€“ x = 9 â€“ 1 = 8 which is even â‡’ 8 âˆ‰ y

For x = 2:

y â€“ x = 4 â€“ 2 = 2 which is even â‡’ 4 âˆ‰ y

y â€“ x = 6 â€“ 2 = 4 which is even â‡’ 6 âˆ‰ y

y â€“ x = 9 â€“ 2 = 7 which is odd â‡’ 8 âˆˆ y

For x = 3:

y â€“ x = 4 â€“ 3 = 1 which is odd â‡’ 4 âˆˆ y

y â€“ x = 6 â€“ 3 = 3 which is odd â‡’ 6 âˆˆ y

y â€“ x = 9 â€“ 3 = 6 which is even â‡’ 8 âˆ‰ y

For x = 5:

x â€“ y = 5 â€“ 4 = 1 which is odd â‡’ 4 âˆˆ y

y â€“ x = 6 â€“ 5 = 1 which is odd â‡’ 6 âˆˆ y

y â€“ x = 9 â€“ 5 = 4 which is even â‡’ 8 âˆ‰ y

âˆ´ R = {(1, 4), (1, 6), (2, 8), (3, 4), (3, 6), (5, 4), (5, 6)}

Domain of relation R = {1, 2, 3, 5}

Range of relation R = {4, 6, 8}

### Question 17. Write the relation R = {(x, x^{3}): x is a prime number less than 10} in roster form.

**Solution:**

Given:R = {(x, x^{3}): x is a prime number less than 10}Prime numbers less than 10 are 2, 3, 5 and 7

âˆ´ R = {(2, 2

^{3}), (3, 3^{3}), (5, 5^{3}), (7, 7^{3})}R = {(2, 8), (3, 27), (5, 125), (7, 343)}

So,

Domain of relation R = {2, 3, 5, 7}

Range of relation R = {8, 27, 125, 343}

### Question 18. Let A = {1, 2, 3, 4, 5, 6}. Let R be a relation on A defined by

### R = {(a, b): a, b âˆˆ A, b is exactly divisible by a}

### (i) Write R in roster form

### (ii) Find the domain of R

### (iii) Find the range of R.

**Solution:**

Given:R= {(a, b): a, b âˆˆ A, b is exactly divisible by a}

A= {1, 2, 3, 4, 5, 6}

Here,

6 is exactly divisible by 1, 2, 3 and 6

5 is exactly divisible by 1 and 5

4 is exactly divisible by 1, 2 and 4

3 is exactly divisible by 1 and 3

2 is exactly divisible by 1 and 2

1 is exactly divisible by 1

(i)R = {(1, 1), (2, 1), (2, 2), (3, 1), (3, 3), (4, 1), (4, 2),(4, 4), (5, 1), (5, 5), (6, 1), (6, 2), (6, 3), (6, 6)}

(ii)Domain of relation R = {1, 2, 3, 4, 5, 6}

(iii)Range of relation R = {1, 2, 3, 4, 5, 6}

### Question 19. The figure shows a relationship between the sets P and Q. Write this relation in

### (i) Set builder form

### (ii) Roster form

### (iii) What is its domain and range?

**Solution:**

(i)Since 5 â€“ 3 = 6 â€“ 4 = 7 â€“ 5 = 2âˆ´ x â€“ y = 2 where x âˆˆ P and y âˆˆ Q

So,

R = {(x, y): x â€“ y = 2, x âˆˆ P, y âˆˆ Q}

(ii)Now, R = {(5, 3), (6, 4), (7, 5)}

(iii)Domain of relation R = {5, 6, 7}Range of relation R = {3, 4, 5}

### Question 20: Let R be the relation on Z defined by R = {(a, b) Z, a â€“ b is an integer}. Find the domain and range of R.

**Solution:**

Given:R = {(a, b) âˆˆ Z, a â€“ b is an integer}Z denotes integer, and here a and b both are integers

We know that difference of two integers is always an integer

âˆ´ a and b can be any integer in relation R

The domain of relation R = Z (as a âˆˆ Z)

The range of relation R = Z (as b âˆˆ Z)

### Question 21: For the relation R_{1} defined on R by the rule (a, b) âˆˆ R_{1} <=>1 + ab > 0. Prove that: (a, b) âˆˆ R_{1} and (b, c) âˆˆ R_{1} â‡’(a, c) âˆˆ R_{1} is not true for all a, b, c âˆˆ R.

**Solution:**

To prove:(a, b) âˆˆ R_{1}and (b, c) âˆˆ R_{1}â‡’(a, c)âˆˆ R_{1}is not true for all a, b, c R.

Given:R_{1}= {(a, b)âˆˆ R <=>1 + ab > 0}Let a = 1, b = -0.5, c = -4

Here, (1, -0.5) âˆˆ R

_{1}[âˆµ 1+(1Ã—-0.5) = 0.5 > 0]And, (-0.5, -4) âˆˆ R

_{1}[âˆµ 1+(-0.5Ã—-4) = 3 > 0]But, (1, -4) âˆ‰ R

_{1}[âˆµ 1+(1Ã—-4) = -3 < 0]âˆ´ (a, b) âˆˆ R

_{1}and (b, c) âˆˆ R_{1}â‡’(a, c) R_{1}is not true for all a, b, c âˆˆ RHence, Proved.

Note: Here, R

_{1}is a relation whereas R denotes a real number.

### Question 22. Let R be a relation on N x N defined by (a, b) R (c, d)<=> a + d = b + c for all (a, b), (c, d) âˆˆ N x N.

### Show that:

### i. (a, b) R (a, b) for all (a, b) âˆˆ N x N

### ii. (a, b) R (c, d) â‡’(c, d) R (a, b) for all (a, b), (c, d) âˆˆ N x N

### iii. (a, b) R (c, d) and (c, d) R (e, f) â‡’(a, b) R (e, f) for all (a, b), (c, d), (e, f) âˆˆ N Ã— N

**Solution:**

Given:(a, b) R (c, d) a + d = b + c for all (a, b), (c, d) âˆˆ N x N

(i)(a, b) R (a, b)a + b = b + a for all (a, b) âˆˆ N x N

âˆ´ (a, b) R (a, b) for all (a, b) âˆˆ N x N

(ii)(a, b) R (c, d)a + d = b + c â‡’ c + b = d + a

(c, d) R (a, b) for all (c, d), (a, b) âˆˆ N x N

(iii)(a, b) R (c, d) and (c, d) R (e, f)a + d = b + c and c + f = d + e

a + d + c + f = b + c + d + e

a + f = b + c + d + e â€“ c â€“ d

a + f = b + e

(a, b) R (e, f) for all (a, b), (c, d), (e, f) âˆˆ N Ã— N