Assertion (A) : Let \(\mathbb{Z}\) be the set of integers. A function \(f: \mathbb{Z} \to \mathbb{Z}\) defined as \(f(x)=3x-5, \forall x \in \mathbb{Z}\) is a bijective function.

Reason (R) : A function is a bijective function if it is both surjective and injective.

"\(m\)R\(n\) if and only if \(m\) is a multiple of \(n\), \(m, n\in N\)."

Find whether R is reflexive, symmetric and transitive or not.

A school is organizing a debate competition with participants as speakers \(S = \{S_1, S_2, S_3, S_4\}\) and these are judged by judges \(J = \{J_1, J_2, J_3\}\). Each speaker can be assigned one judge. Let R be a relation from set S to J defined as \(R=\{(x,y)\): speaker \(x\) is judged by judge \(y, x\in S, y\in J\}\).

Based on the above answer the following:

(i) How many relations can be there from S to J? (1 mark)

(ii)A student identifies a function from S to J as \(f=\{(S_{1},J_{1}), (S_2, J_2), (S_{3},J_{2}), (S_{4},J_{3}))\). Check if it is bijective. (1 mark)

(iii)(a) How many one-one functions can be there from set S to set J? (2 marks)

OR

(iii)(b) Another student considers a relation \(R_{1}=\{(S_{1},S_{2}), (S_2, S_4)\}\) in set S. Write minimum ordered pairs to be included in \(R_{1}\) so that \(R_{1}\) is reflexive but not symmetric. (2 marks)

Based on the above answer the following:

(i) How many relations can be there from S to J? (1 mark)

(ii)A student identifies a function from S to J as \(f=\{(S_{1},J_{1}), (S_2, J_2), (S_{3},J_{2}), (S_{4},J_{3}))\). Check if it is bijective. (1 mark)

(iii)(a) How many one-one functions can be there from set S to set J? (2 marks)

OR

(iii)(b) Another student considers a relation \(R_{1}=\{(S_{1},S_{2}), (S_2, S_4)\}\) in set S. Write minimum ordered pairs to be included in \(R_{1}\) so that \(R_{1}\) is reflexive but not symmetric. (2 marks)