Sets, Relations & Functions
Sets and Relations
nta_pyq_2025_jan
Grade 11
Question:
Let X = R \times R. Define a relation R on X as : (a1 , b1 ) R (a2 , b2 ) \Leftrightarrow b1 = b2 Statement I : R is an equivalence relation. Statement II : For some (a, b) \in X, the set S = {(x, y) \in X : (x, y)R(a, b)} represents a line parallel to y = x. In the light of the above statements, choose the correct answer from the options given below :
Both Statement I and Statement II are false
Statement I is true but Statement II is false
Both Statement I and Statement II are true
Statement I is false but Statement II is true
Step-by-Step Solution
Key Concept: Apply the core result for equivalence relations and set operations and simplify using the given constraints.
Reflexive : (a , b) R (a , b ) \Rightarrow b = b 1 1 1 1 1 True (2) Symmetric : (a1 , b1 ) R (a2 , b2 ) \Rightarrow b1 = b2 } True (a2 , b2 ) R (a1 , b1 ) \Rightarrow b2 = b1 Transitive : (a1 , b1 ) R (a2 , b2 ) \Rightarrow b1 = b2 & (a2 , b2 ) R (a3 , b3 ) b2 = b3 \Rightarrow (a1 , b1 ) R (a3 . b3 ) \Rightarrow True }b1 = b3 Hence Relation R is an equivence relation Statement-I is true. For statement -II \Rightarrow y = b so False
Correct Answer: 2