Class 12 Relation and Function Maths Question
CHAPTER 1 - RELATIONS & FUNCTIONS
1) Let Z be the set of integers and R be a relation defined in Z such that
aRb if (a - b) is divisible by 5. Then R partitions the set Z into ______ pairwise disjoint subsets.
2) Consider set A = {1, 2, 3} and the relation R = {(1, 2)}, then R is a transitive relation. State true or false.
3) Every relation which is symmetric and transitive is reflexive also.
State true or false.
4) Let R be a relation in set N, given by R = {(a, b) : a = b - 2, b > 6} then (3, 8) ∈ R. State true or false with reason.
5) If a relation defined as R = {(x,x), (y,y), (z,z), (x.z)} in set A = {x,y,z} then R is _____________ (reflexive symmetric) relation.
6) For the set A = {1, 2, 3}, define a relation R in the set A as follows: R = {(1, 1), (2, 2), (3, 3), (1, 3)}. Write the ordered pairs to be added to R to make it the smallest equivalence relation.
7) If R = {(x, y) : x + 2y = 8} is a relation on N, write the range of R.
8) Let R = {(a, a²): a is a prime number less than 5} be a relation. Find the range of R.
9) A reflexive relation is identity relation also. State true or false.
10) Let set A = {1, 2, 3}, define relation R on A as R = {(a, b) ∈ A × A : a + b < 6}. Show that R is a universal relation.
11) Let A = {a, b, c}, find the total number of distinct relations in set A.
12) For any relation R in a set A, we can define the inverse relation R⁻¹ by a R⁻¹ b if and only if bRa. Prove that R is symmetric if and only if R = R⁻¹.
✤ Long Answer I / Long Answer II Type
13) Prove that the relation R in the set A = {5, 6, 7, 8, 9} given by R = {(a, b) : |a - b| is divisible by 2}, is an equivalence relation. Find all elements related to the element 6. [Foreign 2013]
14) Let P be the set of all the points in a plane and the relation R in set P be defined as R = {(A, B) ∈ P × P | distance between points A and B is less than 3 units}. Show that the relation R is not an equivalence relation.
15) Let f: A → A be a given function. A relation R in set A is given by R = {(a, b) ∈ A × A | f(a) = f(b)}. Check, whether R is an equivalence relation.
16) Prove that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a - b| is even}, is an equivalence relation. Show that all the elements {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to
any element of {2, 4}. [NCERT; DoE; Chennai 2015]
17. Check whether the relation R defined in the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.
[NCERT]
18. Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance
of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the
circle passing through P with origin as centre.
[NCERT]
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