JEE 2022 — Mathematics PYQ
JEE | Mathematics | 2022Let a set A=A1∪A2∪⋯∪Ak, where Ai∩Aj=ϕ for i=j,1≤i,j≤k. Define the relation R from A to A by R={(x,y):y∈Ai if and only if x∈Ai,1≤i≤k}. Then, R is:
Choose the correct answer:
- A.
reflexive, symmetric but not transitive
- B.
reflexive, transitive but not symmetric
- C.
reflexive but not symmetric and transitive
- D.
an equivalence relation
(Correct Answer)
an equivalence relation
Explanation
Solution
Reflexive: Since x∈Ai⟺x∈Ai is always true, (x,x)∈R for all x∈A.
Symmetric: If (x,y)∈R, then both x,y∈Ai. This implies (y,x)∈R as well.
Transitive: If (x,y)∈R and (y,z)∈R, then x,y,z all belong to the same subset Ai. Thus, (x,z)∈R.
Since R satisfies all three properties, it is an equivalence relation.
Correct Option: (D)
Explanation
Solution
Reflexive: Since x∈Ai⟺x∈Ai is always true, (x,x)∈R for all x∈A.
Symmetric: If (x,y)∈R, then both x,y∈Ai. This implies (y,x)∈R as well.
Transitive: If (x,y)∈R and (y,z)∈R, then x,y,z all belong to the same subset Ai. Thus, (x,z)∈R.
Since R satisfies all three properties, it is an equivalence relation.
Correct Option: (D)