**Q.1: Let X = {1, 2, 3, 4, . . . . . 14}. Define a relation Z from X to X by Z= {(a, b): 3a – b = 0, where a, b ∈ X}. Find its co – domain, domain and range.**

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**Sol:**

**The relation ‘Z’ from ‘X to X’ is:**

Z = {(a, b): 3a – b = 0, where a, b

Z = {(a, b): 3a = b, where a, b

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

The **domain** **of Z** is the set of all the first elements of the ordered pairs in the relation.

**Domain of Z = {1, 2, 3, 4}**

The set X is the co – domain of the relation Z.

Therefore, co – domain of Z = X = {1, 2, 3, 4, . . . . . . 14}

The range of Z is the set of the second elements of the ordered pairs in the relation.

**Therefore, Range of Z = {3, 6, 9, 12}**

**Q.2: Define a relation Z on the set N of natural no. by Z = {(a, b): b = a + 5, a is a natural no less than 4; a, b ∈ N}. Give this relationship in the roaster form. Find the domain and the range.**

**Sol:**

Z = {(a, b): b = a + 5, a is a natural number less than 4; a, b

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

**Z = {(1, 6), (2, 7), (3, 8)}**

The domain of Z is the set of all the first elements of the ordered pairs in the relation.

**Domain of Z = {1, 2, 3}**

The **range of Z** is the set of the second elements of the ordered pairs in the relation.

**Therefore, Range of Z = {6, 7, 8}**

**Q.3: M = {1, 2, 3, 5} and N = {4, 6, 9}. Define a relation Z from M to N by Z = {(a, b): the difference between a and b is odd; a ∈ M, b ∈ N}. Find Z in roster form.**

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**Sol:**

**M = {1, 2, 3, 5}**

**N = {4, 6, 9}**

Z = {(a, b): the difference between a and b is odd; a

**Therefore, Z = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}**

**Q.4: The figure given below shows a relationship between the sets A and B. Find the following relation:**

**(i) In set-builder form**

**(ii) In roster form.**

**What is its range and domain?**

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**Sol:**

According to the information given in the figure:

**A = {5, 6, 7}**

**B = {3, 4, 5}**

**(i).** Z = {(a, b): b = a – 2; a

(or), **Z = {(a, b): b = a – 2 for a = 5, 6, 7}**

**(ii).** Z = {(5, 3), (6, 4), (7, 5)}

**Domain of Z = {5, 6, 7}**

**Range of Z = {3, 4, 5}**

**Q.5: Let X = {1, 2, 3, 4, 6}. Let Z be the relation on X defined by {(p, q): p, q ∈ X, q is divisible by p}.**

**(i) Write Z in the roster form**

**(ii) Find domain of Z**

**(iii) Find range of Z**

**Sol:**

X = {1, 2, 3, 4, 6}

Z = {(p, q): p, q

**(i) Z = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}**

**(ii) Domain of Z = {1, 2, 3, 4, 6}**

**(iii) Range of Z = {1, 2, 3, 4, 6}**

**Q.6: Find the range and domain of the relation Z defined by Z = {(a, a + 5): a ∈ {0, 1, 2, 3, 4, 5}}.**

**Sol:**

Z = {(a, a + 5): a

Therefore, Z = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}

**Domain = {0, 1, 2, 3, 4, 5}**

**Range = {5, 6, 7, 8, 9, 10}**

**Q.7: Find the relation Z = {(a, a3): a is a prime number less than 10} in the roster form.**

**Sol:**

Z = {(a,

The prime number less than 10 are **2, 3, 5, and 7**.

Therefore, **Z = {(2, 8), (3, 27), (5, 125), (7, 343)}**

**Q.8: Let X = {a, b, c} and Y = {11, 12}. Find the no. of relations from X to Y.**

**Sol:**

It is given that **X = {a, b, c} **and **Y = {11, 12}.**

X × Y = {(a, 11), (a, 12), (b, 11), (b, 12), (c, 11), (c, 12)}

As n(X × Y) = 6, the no of subsets of X × Y =

**Therefore, the number of relations from X to Y is 26.**

**Q.9: Let Z be the relation on P defined by Z = {(x, y): x, y ∈ P, x – y is an integer}. Find the range and domain of Z.**

**Sol:**

Z = {(x, y): x, y

As we know that the difference between any two integers is always an integer.

**Domain of Z = P**

**Range of Z = P**