Relations and Functions (Class 12)

Let A=\1, 2, 3, 4\ and R=\(1, 2), (2, 3), (1, 4)\ be a relation on A. Let S be the equivalence relation on A such that R S and the number of elements in S is n. Then, the minimum value of n is .

Let A=\1, 2, 3, , 100\. Let R be a relation on A defined by (x, y) R if and only if 2x=3y. Let R1 be a symmetric relation on A such that R R1 and the number of elements in R1 is n. Then, the minimum value of n is .

Let A=\1,2,3, , 7\ and let P(A) denote the power set of A. If the number of functions f: A P(A) such that a f(a), a A is mn, m and n N and m is least, then m+n is equal to .

Let f: R-\0\ R be a function satisfying f ( xy )= f(x)f(y) for all x,y,f(y) 0. If f'(1)=2024, then

1. xf'(x)+2024f(x)=0
2. xf'(x)-2023f(x)=0
3. xf'(x)-2024f(x)=0
4. xf'(x)+f(x)=2024

The number of symmetric relations defined on the set \1,2,3,4\ which are not reflexive is .

Let R be a relation on Z Z defined by (a, b)R(c, d) if and only if ad-bc is divisible by 5. Then R is

1. Reflexive and transitive but not symmetric
2. Reflexive and symmetric but not transitive
3. Reflexive but neither symmetric nor transitive
4. Reflexive, symmetric and transitive

If f(x)= cases2+2x, & -1 x<0\ 1- x3, & 0 x 3 cases; g(x)= cases-x, & -3 x 0\ x, & 0<x 1 cases, then range of (f g)(x) is

1. [0,1)
2. [0,3)
3. (0,1]
4. [0,1]

If R is the smallest equivalence relation on the set \1,2,3,4\ such that \(1,2),(1,3)\ , then the number of elements in R is .

1. 15
2. 10
3. 12
4. 8

Let S=\1,2,3, ,10\. Suppose M is the set of all the subsets of S, then the relation R=\(A, B):A B ; A, B M\ is :

1. symmetric only
2. reflexive only
3. symmetric and reflexive only
4. symmetric and transitive only

The function f:N-\1\ ; defined by f(n)= the highest prime factor of n, is :

1. one-one only
2. neither one-one nor onto
3. onto only
4. both one-one and onto