Question

Read the following passage:

An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1, b2, b3} and G = {g1, g2}, where B represents the set of Boys selected and G the set of Girls selected for the final race.

Based on the above information, answer the following questions:

1. How many relations are possible from B to G? (1)
2. Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
3. Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
   OR
   A function f : B `rightarrow` G be defined by f = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}. Check if f is bijective. Justify your answer. (2)

Answer 1

1. Number of possible relations from B `rightarrow` G
   = `2^(n(B) xx n(G))`
   = 2^{3 × 2}
   = 2⁶
   = 64.
2. 
   
   Every element of set B has two options to map in set G i.e., B₁ can go to G₁ and G₂.
   So, 2 ways (i.e., two functions).
   ∴ Total function = 2 × 2 × 2 = 8
3. R : B `rightarrow` B
   R = {(x, y) : x and y are students of the same sex}
   (b₁, b₁) ∈ R   ...(Reflexive)
   (b₁, b₂) ∈ R `\implies` (b₂, b₁) ∈ R  ...(Symmetric)
   If (b₁, b₂) ∈ R ∧ (b₂, b₃) ∈ R
   `\implies` (b₁, b₃) ∈ R   ...(Transitive)
   `\implies` It is an equivalence relation.
   OR
   Given, B = {b₁, b₂, b₃} and G = {g₁, g₂}
   f = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}
   Since b₁ and b₃ both are related to same element g₁.
   So f is not bijective (one-one).