Question

A school is organizing a debate competition with participants as speakers S = {S1, S2, S3, S4} and these are judged by judges J = {J1, J2, J3}. 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 ∈ S, y ∈ J}.

Based on the above, answer the following:

1. How many relations can be there from S to J?   [1]
2. A student identifies a function from S to J as f = {(S₁, J₁), (S₂, J₂), (S₃, J₂), (S₄, J₃)} Check if it is bijective.   [1]
3. 
   
   1. How many one-one functions can be there from set S to set J?   [2]
                                                   OR
   2. Another student considers a relation R₁ = {(S₁, S₂), (S₂, S₄)} in set S. Write the minimum ordered pairs to be included in R₁ so that R₁ is reflexive but not symmetric.   [2]

Answer 1

(i) Number of relations = 2^mn

= 2^{4 ×3}

= 2¹² 

= 4096

(ii) f = {(S₁, J₁), (S₂, J₂), (S₃, J₂), (S₄, J₃)} is not one-one as f(S₂) = f(S₃) = J₂ but S₂ ≠ S₃

∴ Not bijective

(iii) (a) Number of one-one function ⁿP_m

= ⁴P₃

= `(4!)/(1!)`

= 24

OR

(iii) (b) R₁= {(S₁, S₂), (S₂, S₄)}

To make it reflexive but not symmetric, add (S₁, S₁), (S₂, S₂), (S₄, S₄).

∴ Minimum number of ordered pairs = 3