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Exercise - 2.2

Question-1 :-  Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

Solution :-
   Give that A = {1, 2, 3,...,14} and R = {(x, y) : 3x – y = 0, where x, y ∈ A}.
   R = {(1, 3), (2, 6), (3, 9), (4, 12)}
   Domain of R = {1, 2, 3, 4}
   Range of R = {3, 6, 9, 12}
   Co-domain of R = {1, 2, 3,...,14}
   

Question-2 :-  Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range.

Solution :-
   Given that the set N of natural numbers by R = {(x, y) : y =  x + 5, x is a natural number less than 4; x, y ∈ N}. 
   R = {(1, 6), (2, 7), (3, 8)}
   Domain of R = {1, 2, 3}
   Range of R = {6, 7, 8}
   Co-domain of R = the set N of natural numbers.
   

Question-3 :-  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 ∈ A, y ∈ B}. Write R in roster form.

Solution :-
   Given that A = {1, 2, 3, 5} and B = {4, 6, 9}
   R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B} 
   R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}
    

Question-4 :-  The given Figure shows a relationship between the sets P and Q. Write this relation
(i) in set-builder form
(ii) roster form. What is its domain and range? relation

Solution :-
    Give that, P = {5, 6, 7}, Q = {3, 4, 5}
(i)	R = {(x, y): y = x – 2; x ∈ P} 
    R = {(x, y): y = x – 2 for x = 5, 6, 7} 

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

Question-5 :-  Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b): a , b ∈ 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 that A = {1, 2, 3, 4, 6} and R = {(a, b): a , b ∈ A, b is exactly divisible by a}.
(i) R = {(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 R = {1, 2, 3, 4, 6} 
(iii) Range of R = {1, 2, 3, 4, 6} 
   

Question-6 :-  Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.

Solution :-
   Given that R = {(x, x + 5): x ∈ {0, 1, 2, 3, 4, 5}}
   R = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} 
   Domain of R = {0, 1, 2, 3, 4, 5}
   Range of R = {5, 6, 7, 8, 9, 10}.
    

Question-7 :-  Write the relation R = {(x, x³) : x is a prime number less than 10} in roster form.

Solution :-
   Given that R = {(x, x³): x is a prime number less than 10}.
   The prime numbers less than 10 are 2, 3, 5, and 7. 
   R = {(2, 8), (3, 27), (5, 125), (7, 343)}
   

Question-8 :-  Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.

Solution :-
   Given that A = {x, y, z} and B = {1, 2}.
   A × B = {(x, 1), (x, 2), (y, 1), (y, 2), (z, 1), (z, 2)}.
   Since n(A × B) = 6, the number of subsets of A × B is 2⁶.
   Therefore, the number of relations from A to B is 2⁶.
   

Question-9 :-  Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R.

Solution :-
   Given that R = {(a, b): a,  b ∈ Z, a – b is an  integer}.
   It is known that the difference between any two integers is always an integer. 
   Domain of R = Z
   Range of R = Z
    
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