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Domain and Range of a Relation
Solved examples on domain and range of a relation:1. In the given ordered pair (4, 6); (8, 4); (4, 4); (9, 11); (6, 3); (3, 0); (2, 3) find the following relations. Also, find the domain and range.(a) Is two less than (b) Is less than (c) Is greater than (d) Is equal to Solution: (a) R_{1} is the set of all ordered pairs whose 1^{st} component is two less than the 2^{nd} component. Therefore, R_{1} = {(4, 6); (9, 11)} Also, Domain (R_{1}) = Set of all first components of R_{1} = {4, 9} and Range (R_{2}) = Set of all second components of R_{2} = {6, 11} (b) R_{2} is the set of all ordered pairs whose 1^{st} component is less than the second component. Therefore, R_{2} = {(4, 6); (9, 11); (2, 3)}. Also, Domain (R_{2}) = {4, 9, 2} and Range (R_{2}) = {6, 11, 3} (c) R_{3} is the set of all ordered pairs whose 1^{st} component is greater than the second component. Therefore, R_{3} = {(8, 4); (6, 3); (3, 0)} Also, Domain (R_{3}) = {8, 6, 3} and Range (R_{3}) = {4, 3, 0} (d) R_{4} is the set of all ordered pairs whose 1^{st} component is equal to the second component. Therefore, R_{4} = {(3, 3)} Also, Domain (R) = {3} and Range (R) = {3} 2. Let A = {2, 3, 4, 5} and B = {8, 9, 10, 11}. Let R be the relation ‘is factor of’ from A to B. (a) Write R in the roster form. Also, find Domain and Range of R. (b) Draw an arrow diagram to represent the relation. Solution: (a) Clearly, R consists of elements (a, b) where a is a factor of b. Therefore, Relation (R) in the roster form is R = {(2, 8); (2, 10); (3, 9); (4, 8), (5, 10)} Therefore, Domain (R) = Set of all first components of R = {2, 3, 4, 5} and Range (R) = Set of all second components of R = {8, 10, 9} (b) The arrow diagram representing R is as follows:
Solution: Workedout problems on domain and range of a relation:4. Let A = {1, 2, 3, 4, 5} and B = {p, q, r, s}. Let R be a relation from A in B defined by R = {1, p}, (1, r), (3, p), (4, q), (5, s), (3, p)} Find domain and range of R. Solution: Given R = {(1, p), (1, r), (4, q), (5, s)} Domain of R = set of first components of all elements of R = {1, 3, 4, 5} Range of R = set of second components of all elements of R = {p, r, q, s} 5. Determine the domain and range of the relation R defined by R = {x + 2, x + 3} : x ∈ {0, 1, 2, 3, 4, 5} Solution: Since, x = {0, 1, 2, 3, 4, 5} Therefore, x = 0 ⇒ x + 2 = 0 + 2 = 2 and x + 3 = 0 + 3 = 3 x = 1 ⇒ x + 2 = 1 + 2 = 3 and x + 3 = 1 + 3 = 4 x = 2 ⇒ x + 2 = 2 + 2 = 4 and x + 3 = 2 + 3 = 5 x = 3 ⇒ x + 2 = 3 + 2 = 5 and x + 3 = 3 + 3 = 6 x = 4 ⇒ x + 2 = 4 + 2 = 6 and x + 3 = 4 + 3 = 7 x = 5 ⇒ x + 2 = 5 + 2 = 7 and x + 3 = 5 + 3 = 8 Hence, R = {(2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8)} Therefore, Domain of R = {a : (a, b) ∈R} = Set of first components of all ordered pair belonging to R. Therefore, Domain of R = {2, 3, 4, 5, 6, 7} Range of R = {b : (a, b) ∈ R} = Set of second components of all ordered pairs belonging to R. Therefore, Range of R = {3, 4, 5, 6, 7, 8} 6. Let A = {3, 4, 5, 6, 7, 8}. Define a relation R from A to A by R = {(x, y) : y = x  1}. • Depict this relation using an arrow diagram. • Write down the domain and range of R. Solution: 7. The adjoining figure shows a relation between the sets A and B. Solution: 7th Grade Math Problems


