Class 12 Relations And Functions · Case Study4 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Case Study #196
Let I be a set of eligible citizens and V a nonempty proper subset consisting of those who voted. Define R on I by xRy iff both x and y voted, so R=V×V.
Part (1) 1 Marks
X voted but Y did not. Is XRY? Explain.
Answer: No.
- Membership in R requires both citizens to belong to V.
- Y∉V, so (X,Y)∉R and XRY is false.
Part (2) 1 Marks
X and spouse W both voted. Do XRW and WRX hold?
Answer: Both hold.
- X,W∈V, so both (X,W) and (W,X) belong to V×V.
- Thus XRW and WRX are true.
Part (3) 2 Marks
Three friends F₁,F₂,F₃ voted. If F₁RF₂ and F₂RF₃, does F₁RF₃ hold? Explain.
Answer: Yes; the relation is transitive.
- The hypotheses imply F₁,F₂,F₃∈V.
- In particular F₁,F₃∈V, so (F₁,F₃)∈V×V. This argument holds for any such triple.
- Therefore F₁RF₃ holds and R is transitive.
Part (4) 2 Marks
Alternatively, if Shyam voted, find the set {y∈I:Shyam Ry}. Explain when this can be called his equivalence class.
Answer: The related set is V; it is an equivalence class for R restricted to V.
- Shyam∈V and Shyam Ry holds precisely when y∈V.
- Thus the related set is V. On all of I, R is not reflexive because a nonvoter has no diagonal pair; it is not an equivalence relation there.
- Restricted to V, R=V×V is an equivalence relation and Shyam’s equivalence class is V.