Sets, Relations & Functions
Sets and Relations
nta_pyq_2025_jan
Grade 11
Question:
The relation R = {(x, y) : x, y \in Z and x + y is even } is:
reflexive and symmetric but not transitive
an equivalence relation
symmetric and transitive but not reflexive
reflexive and transitive but not symmetric
Step-by-Step Solution
Key Concept: Apply the core result for equivalence relations and set operations and simplify using the given constraints.
For reflexive (x, x) \in R, x \in Z (2) \Rightarrow x + x = 2x \to even For symmetric of (x, y) \in R then (y, x) \in R when x, y \in Z x + y \to even \Rightarrow y + x \to even for transitive if (x, y) \in R \Rightarrow x + y \to even (y, z) \in R \Rightarrow y + z \to even x + 2y + z \to even \Rightarrow x + z is even \Rightarrow (x, z) \in R \Rightarrow R is an equivalence relation.
Correct Answer: 2