RELATIONS AND FUNCTIONS Maths - RELATIONS AND FUNCTIONS

R is a relation on the set Z of integers and it is given by (x, y) ∈ R ⇔ | x - y | ≤ 1. Then, R is:

1. Reflexive and transitive.
2. Reflexive and symmetric.
3. Symmetric and transitive.
4. An equivalence relation.

The number of binary operation that can be defined on a set of 2 elements is:

1. 8
2. 4
3. 16
4. 64

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.

1. R is reflexive and symmetric but not transitive.
2. R is reflexive and transitive but not symmetric.
3. R is symmetric and transitive but not reflexive.
4. R is an equivalence relation.

Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: If X = {0, 1, 2} and the function defined by f(x) = x² - 2 is surjection then Y = {-2, -1, 0}.
Reason: If f : X → Y is surjective if f(X) = Y.

1. Both A and R are true and R is the correct explanation of A. 
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.
5. Both A and R are fals.

If R = {(x, y) : x + 2y = 8} is a relation on N, write the range of R.

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.

If f : R → R be defined by f(x) = (3 - x³)^1/3, then find fof(x).

If f : R → R is defined by f(x) = x², find f^-1(-25).

The binary operation *: R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.

Symmetric and transitive but not reflexive.

Let f : R → R be defined as f(x) = 10x + 7. Find the function g : R → R such that gof = fog = I_R.