Class 12th Relations and Functions Practice sheet 2
Ch-1 Relations and Functions
DPP02
Q 1. Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) R = {(x,y) : x and y work at the same place }
(ii) R = {(x, y): x and y live in the same locality}
(iii) R = {(x, y): x is wife of y}
(iv) R = {(x,y): x is father of y} [NCERT]
Q 2. Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows: R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 ={(a, a)} R3 = {(b, c)} R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is
(i) reflexive (ii) symmetric (iii) transitive.
Q 3. Test whether the following relations R1, R2, and R3 are (i) reflexive (ii) symmetric and (iii) transitive:
(i) R1 on Q0 defined by (a,b) belongs to R1 : a = 1/b.
(ii) R2 on Z defined by (a, b) belongs to R2 : | a – b | ≤ 5
(iii) R3 on R defined by (a, b) belongs to R3 & a2 – 4ab + 3b2 = 0.
Q 4. Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2= {(2, 2), (3,1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is
(i) reflexive (ii) symmetric (iii) transitive.
Q 5. The following relations are defined on the set of real numbers.
(i) aRb if a – b > 0 (ii) aRb iff 1 + ab > 0 (iii) aRb if | a | ≤ b.
Find whether these relations are reflexive, symmetric or transitive.
Q 6. Check whether the relation R defined on the set A = {1,2,3,4,5,6} as R = {(a, b):b = a + 1} is reflexive, symmetric or transitive. [NCERT]
Q 7. Check whether the relation R on R defined by R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive.[NCERT, CBSE 2010]
Q 8. Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.
Q 9. If A = {1, 2, 3,4} define relations on A which have properties of being
(i) reflexive, transitive but not symmetric (ii) symmetric but neither reflexive nor transitive (iii) reflexive, symmetric and transitive.
Q 10. Let R be a relation defined on the set of natural numbers N asR = {(x, y): x, y belongs to N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.
Q 11. Is it true that every relation which is symmetric and transitive is also reflexive ? Give reasons.
Q 12. An integer m is said to be related to another integer n if m is a multiple of n. Check if the relation is symmetric, reflexive and transitive.
Q 13. Show that the relation ‘>=’ on the set R of all real numbers is reflexive and transitive but not symmetric.
Q 14. Given the relation R = {(1,2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.
Q 15. Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive reflection on A.