Class 12 relations and functions important questions

1. Let \( R \) be a relation on the set \( \mathbb{N} \times \mathbb{N} \) defined by \( (a, b)R(c, d) \) if and only if \( ad(b+c) = bc(a+d) \). Show that \( R \) is an equivalence relation.

2. Let \( f: \mathbb{R} \to \mathbb{R} \) be defined as:

\[ f(x) = \begin{cases} x+1 & \text{if } x \in \mathbb{Q} \\ x-1 & \text{if } x \notin \mathbb{Q} \end{cases} \]

Show that \( f \) is a bijective function and find \( f \circ f(x) \).

3. Let \( A = \{1, 2, 3, \dots, n\} \) and \( B = \{a, b\} \). Determine the number of onto functions from \( A \) to \( B \). Also, find the total number of relations from \( A \) to \( B \) that are not functions.

4. Consider the set \( A = \{x \in \mathbb{Z} : 0 \le x \le 12\} \). Let \( R \) be the relation on \( A \) defined by \( R = \{(a, b) : |a-b| \text{ is a multiple of } 4\} \). Find the set of all elements related to 1. Also, write all distinct equivalence classes.

5. Let \( m \) be a fixed positive integer. Define a relation \( R \) on \( \mathbb{Z} \) as \( xRy \) if and only if \( (x-y) \) is divisible by \( m \). If \( m=5 \), find the value of \( [2] \cap [7] \) where \( [a] \) denotes the equivalence class of \( a \).

6. If the function \( f: [1, \infty) \to [1, \infty) \) is defined by \( f(x) = 2^{x(x-1)} \), find \( f^{-1}(x) \).

7. Consider \( f: \mathbb{R} \to (-1, 1) \) defined by \( f(x) = \frac{10^x – 10^{-x}}{10^x + 10^{-x}} \). Show that \( f \) is invertible and find its inverse.

8. Give an example of a relation \( R \) on a set \( A = \{1, 2, 3\} \) which is symmetric and transitive but not reflexive. Justify your answer.

9. Let \( f(x) = \frac{x}{\sqrt{1+x^2}} \). Show that \( (f \circ f \circ f)(x) = \frac{x}{\sqrt{1+3x^2}} \). Hence, generalize the formula for \( f_n(x) = (f \circ f \circ \dots \circ f)(x) \) (\( n \) times).

10. Let \( X \) be a non-empty set and \( P(X) \) be its power set. Define a relation \( R \) in \( P(X) \) as: \( ARB \) iff \( A \cap B = A \) for all \( A, B \in P(X) \). Is \( R \) an equivalence relation? Justify.

11. Let \( A \) be the set of all students of a school. Let \( R \) be a relation on \( A \) defined by \( R = \{(a, b) : \text{age of } a \text{ and } b \text{ differ by at most 3 months}\} \). Determine if \( R \) is reflexive, symmetric, and transitive.

12. Let \( f: \mathbb{R} \setminus \{-1\} \to \mathbb{R} \setminus \{1\} \) be given by \( f(x) = \frac{x-2}{x+1} \). Is \( f \) a bijection? If so, find \( f^{-1} \).

13. Let \( f: \mathbb{R} \to \mathbb{R} \) be the Signum function and \( g: \mathbb{R} \to \mathbb{R} \) be the Greatest Integer Function. Does \( fog \) and \( gof \) coincide on the interval \( (0, 1] \)? Explain.

14. If \( f(x) = \log\left(\frac{1+x}{1-x}\right) \) and \( g(x) = \frac{3x+x^3}{1+3x^2} \), then show that \( (f \circ g)(x) = 3f(x) \).

15. Let \( R \) be a relation on \( \mathbb{N} \times \mathbb{N} \) defined by \( (a, b)R(c, d) \) iff \( a+d = b+c \). Find the equivalence class of \( (2, 5) \).

16. If \( A = \{1, 2, 3, 4\} \), find the maximum and minimum number of elements an equivalence relation on \( A \) can have.

17. Find the domain and range of the function \( f(x) = \sqrt{\log_{10} \left( \frac{5x-x^2}{4} \right)} \).

18. Let \( f: [2, \infty) \to \mathbb{R} \) be defined by \( f(x) = x^2 – 4x + 5 \). Show that \( f \) is one-one but not onto. How can you redefine the codomain to make it a bijection? Find the inverse in that case.

19. Let \( f: \mathbb{N} \to \mathbb{N} \) be defined by \( f(n) = \frac{n+1}{2} \) if \( n \) is odd and \( f(n) = \frac{n}{2} \) if \( n \) is even for all \( n \in \mathbb{N} \). State whether the function \( f \) is bijective. Justify.

20. Show that the function \( f: \mathbb{R} \to \{x \in \mathbb{R} : -1 < x < 1\} \) defined by \( f(x) = \frac{x}{1+|x|} \) is one-one and onto.