class 12th Relations and Functions Practice Sheet 1
Ch-1 Relations and Functions
DPP01
Q 1. Three relations R1, R2 and R3 are defined on set A = {a, b, c} as follows :
(i) R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
(ii) R2 = {(a, a), (b, a), (a, c), (c, a)}
(iii) R3 = {(a, b), (b, c), (c, a)}.
Find whether each of R1, R2 and R3 is reflexive, symmetric and transitive.
Q 2. Show that the relation R on the set A = {1, 2, 3} given by
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}
is reflective but neither symmetric nor transitive. [NCERT]
Q 3. Show that the relation R on the set A = {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive now transitive. [NCERT]
Q 4. Check the following relations R and S for reflexivity, symmetry and transitivity :
(i) aRb iff b is divisible by a, b belongs to N
(ii) l1Sl2 iff l1 perpendicular l2, where l1 and l2 are straight lie in a plane.
Q 5. Let a relation R1 on the set R of real numbers be defined as (a, b) belongs to R1 : 1 + ab > 0 for all a, b belongs to R.
Show that R1 is reflexive and symmetric but not transitive.
Q 6. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 be a relation on X given by R1 = {(x, y) : x – y is divisible by 3} and R2 be another relation on X given by R2 = {(x, y) : |x, y| is subset of {1, 4, 7} or {x, y} is subset of {2, 5, 8} or {x, y} is subset of {3, 6, 9}.
Show that R1 = R2. [NCERT]
Q 7. Determine whether each of the following relations are reflective, symmetric and transitive.
(i) Relation R on the set A = {1, 2, 3,………, 13, 14} defined as R = {(x, y) : 3x – y = 0}
(ii) Relation R on the set N of all natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}
(iii) Relation R on the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y) : y is divisible by x}
(iv) Relation R on the set Z of all integer by x} R = {(x, y) : x – y is an integer} [NCERT]
Q 8. Show that the relation R on the set R of all real numbers, defined as R = {(a, b) : a ≤ b2}
is neither reflexive nor symmetric nor transitive. [NCERT]
Q 9. Show that the relation R on R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.
Q 10. Let A = {1, 2, 3}. Then, show that the number of relations containing (1, 2) and (2, 3) which are reflexive and transitive but not symmetric is four. [NCERT]
Q 11. Let S be the set of all points in a plane and R be a relation on S defined as
R = {(P, Q) : Distance between P and Q is less than 2 units}.
Show that R is reflexive and symmetric but not transitive.